Investigation of Crouzeixs Conjecture via Nonsmooth Optimization - - PowerPoint PPT Presentation

1 / 50

Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization

Michael L. Overton Courant Institute of Mathematical Sciences New York University Joint work with Anne Greenbaum, University of Washington and Adrian Lewis, Cornell

January 2017

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

2 / 50

The Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

3 / 50

For A ∈ Cn×n, the field of values (or numerical range) of A is W(A) = {v∗Av : v ∈ Cn, v2 = 1} ⊂ C.

The Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

3 / 50

For A ∈ Cn×n, the field of values (or numerical range) of A is W(A) = {v∗Av : v ∈ Cn, v2 = 1} ⊂ C. Clearly W(A) ⊇ σ(A) where σ denotes spectrum.

The Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

3 / 50

For A ∈ Cn×n, the field of values (or numerical range) of A is W(A) = {v∗Av : v ∈ Cn, v2 = 1} ⊂ C. Clearly W(A) ⊇ σ(A) where σ denotes spectrum. If AA∗ = A∗A, then W(A) = conv σ(A).

The Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

3 / 50

For A ∈ Cn×n, the field of values (or numerical range) of A is W(A) = {v∗Av : v ∈ Cn, v2 = 1} ⊂ C. Clearly W(A) ⊇ σ(A) where σ denotes spectrum. If AA∗ = A∗A, then W(A) = conv σ(A). Toeplitz-Haussdorf Theorem: W(A) is convex for all A ∈ Cn×n.

Examples

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

4 / 50

Let J = 1

• :

W(J) is a disk of radius 0.5

Examples

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

4 / 50

Let J = 1

• :

W(J) is a disk of radius 0.5 B =

• 1

2 −3 4

• :

W(B) is an “elliptical disk”

Examples

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

4 / 50

Let J = 1

• :

W(J) is a disk of radius 0.5 B =

• 1

2 −3 4

• :

W(B) is an “elliptical disk” D = 5 + i 5 − i

• :

W(D) is a line segment

Examples

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

4 / 50

Let J = 1

• :

W(J) is a disk of radius 0.5 B =

• 1

2 −3 4

• :

W(B) is an “elliptical disk” D = 5 + i 5 − i

• :

W(D) is a line segment A = diag(J, B, D) : W(A) = conv (W(J), W(B), W(D))

Example, continued

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

5 / 50

1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 Field of Values of A = diag(J,B,D): J is Jordan block, B full, D diagonal W(A) W(J) W(B) W(D) eig(A)

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A).

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A). The left-hand side is the 2-norm (spectral norm, maximum singular value) of the matrix p(A).

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A). The left-hand side is the 2-norm (spectral norm, maximum singular value) of the matrix p(A). The norm on the right-hand side is the maximum of |p(z)|

• ver z ∈ W(A). By the maximum modulus principle, this must

be attained on bd W(A), the boundary of W(A).

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A). The left-hand side is the 2-norm (spectral norm, maximum singular value) of the matrix p(A). The norm on the right-hand side is the maximum of |p(z)|

• ver z ∈ W(A). By the maximum modulus principle, this must

be attained on bd W(A), the boundary of W(A). If p = χ(A), the characteristic polynomial (or minimal polynomial) of A, then p(A)2 = 0 by Cayley-Hamilton, but pW(A) = 0 only if A = λI for λ ∈ C, so that W(A) = {λ}.

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A). The left-hand side is the 2-norm (spectral norm, maximum singular value) of the matrix p(A). The norm on the right-hand side is the maximum of |p(z)|

• ver z ∈ W(A). By the maximum modulus principle, this must

be attained on bd W(A), the boundary of W(A). If p = χ(A), the characteristic polynomial (or minimal polynomial) of A, then p(A)2 = 0 by Cayley-Hamilton, but pW(A) = 0 only if A = λI for λ ∈ C, so that W(A) = {λ}. If p(z) = z and A is a 2 × 2 Jordan block with 0 on the diagonal, then p(A)2 = 1 and W(A) is a disk centered at 0 with radius 0.5, so the left and right-hand sides are equal.

Crouzeix’s Conjecture

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

6 / 50

Let p = p(z) be a polynomial and let A be a square matrix.

• M. Crouzeix conjectured in “Bounds for analytical functions of

matrices”, Int. Eq. Oper. Theory 48 (2004), that for all p and A, p(A)2 ≤ 2 pW(A). The left-hand side is the 2-norm (spectral norm, maximum singular value) of the matrix p(A). The norm on the right-hand side is the maximum of |p(z)|

• ver z ∈ W(A). By the maximum modulus principle, this must

be attained on bd W(A), the boundary of W(A). If p = χ(A), the characteristic polynomial (or minimal polynomial) of A, then p(A)2 = 0 by Cayley-Hamilton, but pW(A) = 0 only if A = λI for λ ∈ C, so that W(A) = {λ}. If p(z) = z and A is a 2 × 2 Jordan block with 0 on the diagonal, then p(A)2 = 1 and W(A) is a disk centered at 0 with radius 0.5, so the left and right-hand sides are equal. Conjecture extends to analytic functions and to Hilbert space

Crouzeix’s Theorem

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

7 / 50

p(A)2 ≤ 11.08 pW(A) i.e., the conjecture is true if we replace 2 by 11.08.

Crouzeix’s Theorem

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

7 / 50

p(A)2 ≤ 11.08 pW(A) i.e., the conjecture is true if we replace 2 by 11.08. “The estimate 11.08 is not optimal. There is no doubt that refinements are possible which would decrease this bound. We are convinced that our estimate is very pessimistic, but to improve it drastically (recall that our conjecture is that 11.08 can be replaced by 2), it is clear that we have to find a completely different method.”

• Michel Crouzeix, “Numerical range and functional

calculus in Hilbert space”, J. Funct. Anal. 244 (2007).

Crouzeix’s Theorem

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

7 / 50

p(A)2 ≤ 11.08 pW(A) i.e., the conjecture is true if we replace 2 by 11.08. “The estimate 11.08 is not optimal. There is no doubt that refinements are possible which would decrease this bound. We are convinced that our estimate is very pessimistic, but to improve it drastically (recall that our conjecture is that 11.08 can be replaced by 2), it is clear that we have to find a completely different method.”

• Michel Crouzeix, “Numerical range and functional

calculus in Hilbert space”, J. Funct. Anal. 244 (2007). Remarkably broad impact: the norm of an analytic function of a matrix A is bounded by a modest constant times its norm on the field of values W(A).

Greatly Improved New Bound from C´ esar Palencia

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

8 / 50

p(A)2 ≤

• 1 +

√ 2

• pW(A)

i.e., the conjecture is true if we replace 2 by 1 + √ 2 Presented at a conference in Greece, summer 2016

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

■

n = 2 (Crouzeix, 2004), and, more generally, the minimum polynomial of A has degree 2 (follows from Tso and Wu, 1999)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

■

n = 2 (Crouzeix, 2004), and, more generally, the minimum polynomial of A has degree 2 (follows from Tso and Wu, 1999)

■

n = 3 and A3 = 0 (Crouzeix, 2013)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

■

n = 2 (Crouzeix, 2004), and, more generally, the minimum polynomial of A has degree 2 (follows from Tso and Wu, 1999)

■

n = 3 and A3 = 0 (Crouzeix, 2013)

■

A is an upper Jordan block with a perturbation in the bottom left corner (Choi and Greenbaum, 2012) or any diagonal scaling of such A (Choi, 2013)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

■

n = 2 (Crouzeix, 2004), and, more generally, the minimum polynomial of A has degree 2 (follows from Tso and Wu, 1999)

■

n = 3 and A3 = 0 (Crouzeix, 2013)

■

A is an upper Jordan block with a perturbation in the bottom left corner (Choi and Greenbaum, 2012) or any diagonal scaling of such A (Choi, 2013)

■

A = TDT −1 with D diagonal and TT −1 ≤ 2 (easy)

Special Cases

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

9 / 50

The conjecture is known to hold for certain restricted classes of polynomials p ∈ P m or matrices A ∈ Cn×n. Let r(A) = maxζ∈W (A) |ζ| (numerical radius) and D = open unit disk

■

p(ζ) = ζm: Am ≤ 2r(Am) ≤ 2r(A)m = 2 maxζ∈W (A) |ζm| (power inequality, Berger 1965, Pearcy 1966)

■

W(A) = D :

• if B ≤ 1, then p(B) ≤ supζ∈D |p(ζ)| (von Neumann, 1951)
• if r(A) ≤ 1, then A = TBT −1 with B ≤ 1 and TT −1 ≤ 2

(Okubo and Ando, 1975), so p(A) ≤ 2p(B)

■

n = 2 (Crouzeix, 2004), and, more generally, the minimum polynomial of A has degree 2 (follows from Tso and Wu, 1999)

■

n = 3 and A3 = 0 (Crouzeix, 2013)

■

A is an upper Jordan block with a perturbation in the bottom left corner (Choi and Greenbaum, 2012) or any diagonal scaling of such A (Choi, 2013)

■

A = TDT −1 with D diagonal and TT −1 ≤ 2 (easy)

■

AA∗ = A∗A (then the constant 2 can be improved to 1).

Computing the Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

10 / 50

The extreme points of a convex set are those that cannot be expressed as a convex combination of two other points in the set.

Computing the Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

10 / 50

The extreme points of a convex set are those that cannot be expressed as a convex combination of two other points in the set. Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed that the extreme points of W(A) can be characterized as ext W(A) = {zθ = v∗

θAvθ : θ ∈ [0, 2π)}

where vθ is a normalized eigenvector corresponding to the largest eigenvalue of the Hermitian matrix Hθ = 1 2

• eiθA + e−iθA∗

.

Computing the Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

10 / 50

The extreme points of a convex set are those that cannot be expressed as a convex combination of two other points in the set. Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed that the extreme points of W(A) can be characterized as ext W(A) = {zθ = v∗

θAvθ : θ ∈ [0, 2π)}

where vθ is a normalized eigenvector corresponding to the largest eigenvalue of the Hermitian matrix Hθ = 1 2

• eiθA + e−iθA∗

. The proof uses a supporting hyperplane argument.

Computing the Field of Values

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

10 / 50

The extreme points of a convex set are those that cannot be expressed as a convex combination of two other points in the set. Based on R. Kippenhahn (1951), C.R. Johnson (1978) observed that the extreme points of W(A) can be characterized as ext W(A) = {zθ = v∗

θAvθ : θ ∈ [0, 2π)}

where vθ is a normalized eigenvector corresponding to the largest eigenvalue of the Hermitian matrix Hθ = 1 2

• eiθA + e−iθA∗

. The proof uses a supporting hyperplane argument. Thus, we can compute as many extreme points as we like. Continuing with the previous example...

Johnson’s Algorithm Finds the Extreme Points

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

11 / 50

−1 1 2 3 4 5 6 −3 −2 −1 1 2 3 θ ∈ [0,0.96] θ ∈ [0.96,2.29] θ ∈ [2.29,3.99] θ ∈ [3.99,5.3] θ ∈ [5.3,2π] The extreme points of W(A) lie in the union of 5 connected sets

Johnson’s Algorithm Finds the Extreme Points

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

11 / 50

−1 1 2 3 4 5 6 −3 −2 −1 1 2 3 θ ∈ [0,0.96] θ ∈ [0.96,2.29] θ ∈ [2.29,3.99] θ ∈ [3.99,5.3] θ ∈ [5.3,2π] The extreme points of W(A) lie in the union of 5 connected sets

But how can we do this accurately, automatically and efficiently?

Chebfun

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

12 / 50

Chebfun (Trefethen et al, 2004–present) represents real- or complex-valued functions on real intervals to machine precision accuracy using Chebyshev interpolation. The necessary degree of the polynomial is determined

• automatically. For example, representing sin(πx) on [−1, 1] to

machine precision requires degree 19. Most Matlab functions are overloaded to work with chebfun’s. Applying Chebfun’s fov to compute the boundary of W(A) for the previous example...

Example, continued

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

13 / 50

1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 Internal points shown are chebfun interpolation points Field of Values of A = diag(J,B,D): J is Jordan block, B full, D diagonal W(A) break points of W(A) ||χ(A)||W(A) attained W(J) W(B) W(D) eig(A)

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 .

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n.

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n. The Crouzeix ratio f is

■

A mapping from Cm+1 × Cn×n to R (associating polynomials p ∈ P m with their vectors of coefficients c ∈ Cm+1 using the monomial basis)

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n. The Crouzeix ratio f is

■

A mapping from Cm+1 × Cn×n to R (associating polynomials p ∈ P m with their vectors of coefficients c ∈ Cm+1 using the monomial basis)

■

Not convex

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n. The Crouzeix ratio f is

■

A mapping from Cm+1 × Cn×n to R (associating polynomials p ∈ P m with their vectors of coefficients c ∈ Cm+1 using the monomial basis)

■

Not convex

■

Not defined if p(A) = 0

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n. The Crouzeix ratio f is

■

A mapping from Cm+1 × Cn×n to R (associating polynomials p ∈ P m with their vectors of coefficients c ∈ Cm+1 using the monomial basis)

■

Not convex

■

Not defined if p(A) = 0

■

Lipschitz continuous at all other points, but not necessarily differentiable

The Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

14 / 50

Define the Crouzeix ratio f(p, A) = pW(A) p(A)2 . The conjecture states that f(p, A) is bounded below by 0.5 independently of the polynomial degree m and the matrix

• rder n. The Crouzeix ratio f is

■

A mapping from Cm+1 × Cn×n to R (associating polynomials p ∈ P m with their vectors of coefficients c ∈ Cm+1 using the monomial basis)

■

Not convex

■

Not defined if p(A) = 0

■

Lipschitz continuous at all other points, but not necessarily differentiable

■

Semialgebraic (its graph is a finite union of sets, each of which is defined by a finite system of polynomial inequalities)

Computing the Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

15 / 50

Numerator: use Chebfun’s fov (modified to return any line segments in the boundary) combined with its overloaded polyval and norm(·,inf).

Computing the Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

15 / 50

Numerator: use Chebfun’s fov (modified to return any line segments in the boundary) combined with its overloaded polyval and norm(·,inf). Denominator: use Matlab’s standard polyvalm and norm(·,2).

Computing the Crouzeix Ratio

Crouzeix’s Conjecture The Field of Values Examples Example, continued Crouzeix’s Conjecture Crouzeix’s Theorem Greatly Improved New Bound from C´ esar Palencia Special Cases Computing the Field

• f Values

Johnson’s Algorithm Finds the Extreme Points Chebfun Example, continued The Crouzeix Ratio Computing the Crouzeix Ratio Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A

15 / 50

Numerator: use Chebfun’s fov (modified to return any line segments in the boundary) combined with its overloaded polyval and norm(·,inf). Denominator: use Matlab’s standard polyvalm and norm(·,2). The main cost is the construction of the chebfun defining the field of values.

Nonsmooth Optimization of the Crouzeix Ratio f

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

16 / 50

Nonsmoothness of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

17 / 50

There are three possible sources of nonsmoothness in f

Nonsmoothness of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

17 / 50

There are three possible sources of nonsmoothness in f

■

When the max value of |p(z)| on bd W(A) is attained at more than one point z (the most important, as this frequently occurs at apparent minimizers)

Nonsmoothness of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

17 / 50

There are three possible sources of nonsmoothness in f

■

When the max value of |p(z)| on bd W(A) is attained at more than one point z (the most important, as this frequently occurs at apparent minimizers)

■

Even if such z is unique, when the normalized vector v for which v∗Av = z is not unique up to a scalar, implying that the maximum eigenvalue of the corresponding Hθ matrix has multiplicity two or more (does not seem to occur at minimizers)

Nonsmoothness of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

17 / 50

There are three possible sources of nonsmoothness in f

■

When the max value of |p(z)| on bd W(A) is attained at more than one point z (the most important, as this frequently occurs at apparent minimizers)

■

Even if such z is unique, when the normalized vector v for which v∗Av = z is not unique up to a scalar, implying that the maximum eigenvalue of the corresponding Hθ matrix has multiplicity two or more (does not seem to occur at minimizers)

■

When the maximum singular value of p(A) has multiplicity two or more (does not seem to occur at minimizers)

Nonsmoothness of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

17 / 50

There are three possible sources of nonsmoothness in f

■

When the max value of |p(z)| on bd W(A) is attained at more than one point z (the most important, as this frequently occurs at apparent minimizers)

■

Even if such z is unique, when the normalized vector v for which v∗Av = z is not unique up to a scalar, implying that the maximum eigenvalue of the corresponding Hθ matrix has multiplicity two or more (does not seem to occur at minimizers)

■

When the maximum singular value of p(A) has multiplicity two or more (does not seem to occur at minimizers) In all of these cases the gradient of f is not defined. But in practice, none of these cases ever occur, except the first

• ne in the limit.

BFGS

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

18 / 50

BFGS (Broyden, Fletcher, Goldfarb and Shanno, all independently in 1970), is the standard quasi-Newton algorithm for minimizing smooth (continuously differentiable) functions.

BFGS

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

18 / 50

BFGS (Broyden, Fletcher, Goldfarb and Shanno, all independently in 1970), is the standard quasi-Newton algorithm for minimizing smooth (continuously differentiable) functions. It works by building an approximation to the Hessian of the function using gradient differences, and has a well known superlinear convergence property under a regularity condition.

BFGS

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

18 / 50

BFGS (Broyden, Fletcher, Goldfarb and Shanno, all independently in 1970), is the standard quasi-Newton algorithm for minimizing smooth (continuously differentiable) functions. It works by building an approximation to the Hessian of the function using gradient differences, and has a well known superlinear convergence property under a regularity condition. Although its global convergence theory is limited to the convex case (Powell, 1976), it generally finds local minimizers efficiently in the nonconvex case too, although there are pathological counterexamples.

BFGS

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

18 / 50

BFGS (Broyden, Fletcher, Goldfarb and Shanno, all independently in 1970), is the standard quasi-Newton algorithm for minimizing smooth (continuously differentiable) functions. It works by building an approximation to the Hessian of the function using gradient differences, and has a well known superlinear convergence property under a regularity condition. Although its global convergence theory is limited to the convex case (Powell, 1976), it generally finds local minimizers efficiently in the nonconvex case too, although there are pathological counterexamples. Remarkably, this property seems to extend to nonsmooth functions too, with a linear rate of local convergence, although the convergence theory is extremely limited (Lewis and Overton, 2013). It builds a very ill conditioned “Hessian” approximation, with “infinitely large” curvature in some directions and finite curvature in other directions.

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS.

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1 ◆ unbounded degree (different method)

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1 ◆ unbounded degree (different method)

■

Optimize over both p with degree ≤ m = n − 1 and A with order n

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1 ◆ unbounded degree (different method)

■

Optimize over both p with degree ≤ m = n − 1 and A with order n

We restrict p to have real coefficients and A to be real, in Hessenberg form (all but one superdiagonal is zero).

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1 ◆ unbounded degree (different method)

■

Optimize over both p with degree ≤ m = n − 1 and A with order n

We restrict p to have real coefficients and A to be real, in Hessenberg form (all but one superdiagonal is zero). We have obtained similar results for p with complex coefficients and complex A (then can take A to be triangular)

Experiments

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Nonsmoothness of the Crouzeix Ratio BFGS Experiments Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

19 / 50

We have run many experiments searching for local minimizers of the Crouzeix ratio using BFGS. Several scenarios:

■

Fix p with degree m, optimize over A with fixed order n ◆ with m = n − 1 ◆ with m = n

■

Fix A with order n, optimize over p with ◆ degree ≤ m = n − 1 ◆ unbounded degree (different method)

■

Optimize over both p with degree ≤ m = n − 1 and A with order n

We restrict p to have real coefficients and A to be real, in Hessenberg form (all but one superdiagonal is zero). We have obtained similar results for p with complex coefficients and complex A (then can take A to be triangular) Subsequent slides show the sorted final values of the Crouzeix ratio after running BFGS for a maximum of 1000 iterations from each of 100 randomly generated starting points.

Fix p, Optimize over A

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

20 / 50

Fix p(z) ≡ zn−1, Optimize over A order n

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

21 / 50

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=3

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=4

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=5

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=6

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=7

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=8

Apparently 0.5, 1 and a few other values are all locally minimal

Final Fields of Values for Lowest Computed f

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

22 / 50

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

n=3

• 2

2

• 3
• 2
• 1

1 2 3

n=4

• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5

n=5

• 4
• 2

2 4

• 4
• 3
• 2
• 1

1 2 3 4

n=6

• 5

5

• 8
• 6
• 4
• 2

2 4 6 8

n=7

• 5

5

• 6
• 4
• 2

2 4 6

n=8

Final Fields of Values for Lowest Computed f

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

22 / 50

• 1
• 0.5

0.5 1

• 1
• 0.5

0.5 1

n=3

• 2

2

• 3
• 2
• 1

1 2 3

n=4

• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5

n=5

• 4
• 2

2 4

• 4
• 3
• 2
• 1

1 2 3 4

n=6

• 5

5

• 8
• 6
• 4
• 2

2 4 6 8

n=7

• 5

5

• 6
• 4
• 2

2 4 6

n=8

Note: eigs(A) → 0 = root(p).

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

23 / 50

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

23 / 50

Independently, Crouzeix and Choi showed that the ratio 0.5 is attained if p(z) = zm and A is the m + 1 by m + 1 matrix

• 2
• if m = 1, or

          √ 2 · 1 · · · · · 1 · √ 2           if m > 1

for which W(A) is the unit disk. We call this the C-matrix.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

23 / 50

Independently, Crouzeix and Choi showed that the ratio 0.5 is attained if p(z) = zm and A is the m + 1 by m + 1 matrix

• 2
• if m = 1, or

          √ 2 · 1 · · · · · 1 · √ 2           if m > 1

for which W(A) is the unit disk. We call this the C-matrix. We conjecture that, when p(z) = zm, this is essentially the only case where 0.5 can be attained (A can be changed via unitary similarity transformations, multiplying A by a scalar, and appending another diagonal block whose field of values is contained in that of the first block).

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

23 / 50

Independently, Crouzeix and Choi showed that the ratio 0.5 is attained if p(z) = zm and A is the m + 1 by m + 1 matrix

• 2
• if m = 1, or

          √ 2 · 1 · · · · · 1 · √ 2           if m > 1

for which W(A) is the unit disk. We call this the C-matrix. We conjecture that, when p(z) = zm, this is essentially the only case where 0.5 can be attained (A can be changed via unitary similarity transformations, multiplying A by a scalar, and appending another diagonal block whose field of values is contained in that of the first block). We base this on the computation of the generalized null space decomposition (staircase decomposition) of the computed A.

(Kublanovskaya 1966, Ruhe 1970, Golub-Wilkinson 1976, Van Dooren 1979, K˚ agstr¨

• m et al, Edelman-Ma 2000, Guglielmi-Overton-Stewart 2015.)

A Local Minimizer with f = 1

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

24 / 50

−1 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 A smooth local min, p(z)=z4 (fixed), dim(A) = 5 (var), ratio = 9.999999999999996e−01 W(A) eigenvalues(A) roots(p)

A Local Minimizer with f = 1

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

24 / 50

−1 1 2 3 4 5 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 A smooth local min, p(z)=z4 (fixed), dim(A) = 5 (var), ratio = 9.999999999999996e−01 W(A) eigenvalues(A) roots(p)

“Ice cream cone” shape

Why is the Crouzeix Ratio One?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

25 / 50

Why is the Crouzeix Ratio One?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

25 / 50

Because for this computed local minimizer, A = U diag(λ, B) U ∗ + E with U unitary, E very small and λ ∈ R, so W(A) ≈ conv(λ, W(B)) with λ active and the block B inactive, that is:

■

pW(A) is attained only at λ

■

|p(λ)| > p(B)2

Why is the Crouzeix Ratio One?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

25 / 50

Because for this computed local minimizer, A = U diag(λ, B) U ∗ + E with U unitary, E very small and λ ∈ R, so W(A) ≈ conv(λ, W(B)) with λ active and the block B inactive, that is:

■

pW(A) is attained only at λ

■

|p(λ)| > p(B)2 So, pW(A) = |p(λ)| = p(A)2 and hence f(p, A) = 1.

Why is the Crouzeix Ratio One?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

25 / 50

Because for this computed local minimizer, A = U diag(λ, B) U ∗ + E with U unitary, E very small and λ ∈ R, so W(A) ≈ conv(λ, W(B)) with λ active and the block B inactive, that is:

■

pW(A) is attained only at λ

■

|p(λ)| > p(B)2 So, pW(A) = |p(λ)| = p(A)2 and hence f(p, A) = 1. Furthermore, the gradient of f(p, ·) is zero at such A, although showing this is more work. Thus, such A is a smooth stationary point of f(zm, ·).

Why is the Crouzeix Ratio One?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

25 / 50

Because for this computed local minimizer, A = U diag(λ, B) U ∗ + E with U unitary, E very small and λ ∈ R, so W(A) ≈ conv(λ, W(B)) with λ active and the block B inactive, that is:

■

pW(A) is attained only at λ

■

|p(λ)| > p(B)2 So, pW(A) = |p(λ)| = p(A)2 and hence f(p, A) = 1. Furthermore, the gradient of f(p, ·) is zero at such A, although showing this is more work. Thus, such A is a smooth stationary point of f(zm, ·). This doesn’t imply that it is a local minimizer, but the numerical results make this evident.

Fix p(z) ≡ zn, Optimize over A with order n

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

26 / 50

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=3

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=4

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=5

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=6

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=7

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=8

Fix p(z) ≡ zn, Optimize over A with order n

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

26 / 50

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=3

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=4

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=5

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=6

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=7

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=8

No value found near 0.5. We conjecture this is not possible.

Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1), Opt. over A (n = 5)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

27 / 50

−10 −5 5 10 −8 −6 −4 −2 2 4 6 8 Best solution found, p(z)=z(z−1)(z2+1) (fixed), dim(A) = 5 (var), ratio = 5.000003853159926e−01 W(A) eigenvalues(A) roots(p)

Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1), Opt. over A (n = 5)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

27 / 50

−10 −5 5 10 −8 −6 −4 −2 2 4 6 8 Best solution found, p(z)=z(z−1)(z2+1) (fixed), dim(A) = 5 (var), ratio = 5.000003853159926e−01 W(A) eigenvalues(A) roots(p)

W(A) is approximately a large disk around all roots of p

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

28 / 50

At first we thought so: we computed f(p, A) agreeing to 0.5 to six digits.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

28 / 50

At first we thought so: we computed f(p, A) agreeing to 0.5 to six digits. However, the closer we get to 0.5, the more W(A) blows up. So, 0.5 is not attained.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

28 / 50

At first we thought so: we computed f(p, A) agreeing to 0.5 to six digits. However, the closer we get to 0.5, the more W(A) blows up. So, 0.5 is not attained. Theorem 1. For any fixed polynomial p of degree m ≥ 1, there exists a divergent sequence {A(k)} of order n = m + 1 for which f(p, A(k)) → 0.5 as k → ∞. Furthermore, we can choose A(k) so {W(A(k))} are disks.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

28 / 50

At first we thought so: we computed f(p, A) agreeing to 0.5 to six digits. However, the closer we get to 0.5, the more W(A) blows up. So, 0.5 is not attained. Theorem 1. For any fixed polynomial p of degree m ≥ 1, there exists a divergent sequence {A(k)} of order n = m + 1 for which f(p, A(k)) → 0.5 as k → ∞. Furthermore, we can choose A(k) so {W(A(k))} are disks. However, 0.5 is not attained.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

28 / 50

At first we thought so: we computed f(p, A) agreeing to 0.5 to six digits. However, the closer we get to 0.5, the more W(A) blows up. So, 0.5 is not attained. Theorem 1. For any fixed polynomial p of degree m ≥ 1, there exists a divergent sequence {A(k)} of order n = m + 1 for which f(p, A(k)) → 0.5 as k → ∞. Furthermore, we can choose A(k) so {W(A(k))} are disks. However, 0.5 is not attained.

• Observation. When we fix p to be any polynomial of degree m

except a monomial, and we optimize over (m + 1) × (m + 1) matrices A, we frequently generate a sequence as described in Theorem 1, except that W(A) are not exactly disks.

Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1), Opt. over A (*n = 4*)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

29 / 50

0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Best solution found, p(z)=z(z−1)(z2+1) (fixed), dim(A) = 4 (var), ratio = 5.000198002813829e−01 W(A) eigenvalues(A) roots(p)

Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1), Opt. over A (*n = 4*)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

29 / 50

0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 Best solution found, p(z)=z(z−1)(z2+1) (fixed), dim(A) = 4 (var), ratio = 5.000198002813829e−01 W(A) eigenvalues(A) roots(p)

W(A) is approximately a tiny disk around some root of p

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

30 / 50

• No. The closer we get to 0.5, the more W(A) shrinks to a point.

In the limit, we get f(p, A) = 0/0 so it is not defined.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

30 / 50

• No. The closer we get to 0.5, the more W(A) shrinks to a point.

In the limit, we get f(p, A) = 0/0 so it is not defined. Theorem 2. Fix p to have degree m with at least two distinct

• roots. Then, for all integers n with 2 ≤ n ≤ m, there exists a

convergent sequence of n × n matrices {A(k)} for which the Crouzeix ratio f(p, A(k)) → 0.5. Furthermore, we can choose A(k) so {W(A(k))} is a sequence of disks shrinking to a root of p.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

30 / 50

• No. The closer we get to 0.5, the more W(A) shrinks to a point.

In the limit, we get f(p, A) = 0/0 so it is not defined. Theorem 2. Fix p to have degree m with at least two distinct

• roots. Then, for all integers n with 2 ≤ n ≤ m, there exists a

convergent sequence of n × n matrices {A(k)} for which the Crouzeix ratio f(p, A(k)) → 0.5. Furthermore, we can choose A(k) so {W(A(k))} is a sequence of disks shrinking to a root of p. However, 0.5 is not attained.

Is the Ratio 0.5 Attained?

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix p(z) ≡ zn−1, Optimize over A

• rder n

Final Fields of Values for Lowest Computed f Is the Ratio 0.5 Attained? A Local Minimizer with f = 1 Why is the Crouzeix Ratio One? Fix p(z) ≡ zn, Optimize over A with order n Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(n = 5) Is the Ratio 0.5 Attained? Fix p(z) ≡ (z − 1)(z − 2)(z2 + 1),

• Opt. over A

(*n = 4*) Is the Ratio 0.5

30 / 50

• No. The closer we get to 0.5, the more W(A) shrinks to a point.

In the limit, we get f(p, A) = 0/0 so it is not defined. Theorem 2. Fix p to have degree m with at least two distinct

• roots. Then, for all integers n with 2 ≤ n ≤ m, there exists a

convergent sequence of n × n matrices {A(k)} for which the Crouzeix ratio f(p, A(k)) → 0.5. Furthermore, we can choose A(k) so {W(A(k))} is a sequence of disks shrinking to a root of p. However, 0.5 is not attained.

• Observation. When we fix p to be any polynomial of degree

m > 1 with at least two roots, and we optimize over m × m matrices A, we sometimes generate a sequence as described in Theorem 2, except that W(A) are not exactly disks.

Fix A, Optimize over p

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

31 / 50

Fix A, Optimize over p

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

32 / 50

If we fix A instead of p, then in general it seems the Crouzeix ratio 0.5 cannot be attained or even approximated by some p.

Fix A, Optimize over p

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

32 / 50

If we fix A instead of p, then in general it seems the Crouzeix ratio 0.5 cannot be attained or even approximated by some p. This seems to be true for all A except when A is essentially a C-matrix.

Fix A, Optimize over p

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

32 / 50

If we fix A instead of p, then in general it seems the Crouzeix ratio 0.5 cannot be attained or even approximated by some p. This seems to be true for all A except when A is essentially a C-matrix. If instead of optimizing over p of fixed maximum degree using BFGS/Chebfun, we use a completely different method,

• ptimizing over all analytic p for fixed A by setting the variables

to the scalars defining a Blaschke product, we get lower Crouzeix ratios than we get using BFGS/Chebfun with fixed maximum degree for p, because we are effectively allowing infinite degree for p, even though the degree of the Blaschke product is fixed to be n − 1 or less.

Fix A, Optimize over p

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks

32 / 50

If we fix A instead of p, then in general it seems the Crouzeix ratio 0.5 cannot be attained or even approximated by some p. This seems to be true for all A except when A is essentially a C-matrix. If instead of optimizing over p of fixed maximum degree using BFGS/Chebfun, we use a completely different method,

• ptimizing over all analytic p for fixed A by setting the variables

to the scalars defining a Blaschke product, we get lower Crouzeix ratios than we get using BFGS/Chebfun with fixed maximum degree for p, because we are effectively allowing infinite degree for p, even though the degree of the Blaschke product is fixed to be n − 1 or less. In the case n = 2, Crouzeix (2004) gives a complete answer: the

• nly A for which optimizing over p gives 0.5 are those for which

W(A) is a disk, and hence A must be a scalar multiple of a Jordan block (a C-matrix since n = 2).

Optimizing over p and A

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

33 / 50

Optimizing over both p (deg ≤ n − 1) and A (order n)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

34 / 50

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=3

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=4

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=5

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=6

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=7

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=8

Optimizing over both p (deg ≤ n − 1) and A (order n)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

34 / 50

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=3

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=4

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=5

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=6

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=7

50 100 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

n=8

Only locally optimal values found are 0.5 and 1

Optimizing over both p and A: Details

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

35 / 50 n f ecc(W(A)) |κ − λ1| |κ − µ1| |κ − µ2| 3 0.500000000000000 2.1e − 08 1.2e − 11 2.2e − 07 2.2e − 07 4 0.500000000000000 1.9e − 04 1.2e − 08 1.7e − 04 1.7e − 04 5 0.500000000000014 3.2e − 04 2.6e − 08 5.0e − 04 5.0e − 04 6 0.500000017156953 8.4e − 02 3.5e − 01 1.7e − 01 3.2e − 01 7 0.500000746246673 1.2e − 01 1.6e − 01 4.4e − 01 1.0e + 00 8 0.500000206563813 1.3e − 01 5.1e − 01 7.2e − 01 7.5e − 01

f is the lowest value f(p, A) found over 100 runs ecc(W(A)) is the eccentricity of W(A) (zero for a disk) κ is the center of W(A) λ1 is the smallest root (in magnitude) of p µ1, µ2 are the two eigenvalues of A that are closest to κ n = 3, 4, 5: two eigenvalues of A and one root of p nearly coincident

Final Fields of Values for Lowest Computed f

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

36 / 50

• 1

1 2

• 2
• 1.5
• 1
• 0.5

0.5 1 1.5 2

n=3

• 1
• 0.5
• 0.8
• 0.6
• 0.4
• 0.2

0.2 0.4 0.6 0.8

n=4

• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5

n=5

• 5

5

• 6
• 4
• 2

2 4 6

n=6

• 5

5

• 6
• 4
• 2

2 4 6

n=7

• 6
• 4
• 2

2 4

• 5

5

n=8

n = 3, 4, 5: two eigenvalues of A and one root of p nearly coincident

An Example: f(p, A) = 0.5000000002

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

37 / 50

An example with m = 4, n = 5: found f = 0.5000000002, with p(z) = −(8.3×10−11)z4−(6.6×10−7)z3+(1.7×10−5)z2+2.6z−1.3 which is nearly linear, with only one moderate sized root: µ = 0.49426, and with A having two eigenvalues 0.492 and 0.497, with mean λ = 0.49424.

An Example: f(p, A) = 0.5000000002

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

37 / 50

An example with m = 4, n = 5: found f = 0.5000000002, with p(z) = −(8.3×10−11)z4−(6.6×10−7)z3+(1.7×10−5)z2+2.6z−1.3 which is nearly linear, with only one moderate sized root: µ = 0.49426, and with A having two eigenvalues 0.492 and 0.497, with mean λ = 0.49424. Using the generalized null space decomposition we find that A − λI = UDU T + E where U is unitary, E ≈ 10−3, D = diag(B1, B2), B1 is a scalar multiple of a 2 × 2 Jordan block (a 2 × 2 C-matrix), and W(B2) ⊂ W(B1).

An Example: f(p, A) = 0.5000000002

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

37 / 50

An example with m = 4, n = 5: found f = 0.5000000002, with p(z) = −(8.3×10−11)z4−(6.6×10−7)z3+(1.7×10−5)z2+2.6z−1.3 which is nearly linear, with only one moderate sized root: µ = 0.49426, and with A having two eigenvalues 0.492 and 0.497, with mean λ = 0.49424. Using the generalized null space decomposition we find that A − λI = UDU T + E where U is unitary, E ≈ 10−3, D = diag(B1, B2), B1 is a scalar multiple of a 2 × 2 Jordan block (a 2 × 2 C-matrix), and W(B2) ⊂ W(B1). Sometimes, we find approximately monomial p with “genuinely” higher degree m, and then A has a similar structure with B1 a scalar multiple of an (m + 1) × (m + 1) C-matrix.

A New Conjecture

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

38 / 50

Based on our experimental results, we conjecture that f(p, A) = 0.5 implies that p(z) = (z − λ)m for some λ and some m, and that A = U diag(B1, B2) U ∗ with U unitary, B1 = λI + tC, where t ∈ C and C is the C-matrix of order m + 1 (zero except the single superdiagonal ( √ 2, 1, . . . , 1, √ 2) or just 2 if m = 1), and W(B2) ⊆ W(B1).

A New Conjecture

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

38 / 50

Based on our experimental results, we conjecture that f(p, A) = 0.5 implies that p(z) = (z − λ)m for some λ and some m, and that A = U diag(B1, B2) U ∗ with U unitary, B1 = λI + tC, where t ∈ C and C is the C-matrix of order m + 1 (zero except the single superdiagonal ( √ 2, 1, . . . , 1, √ 2) or just 2 if m = 1), and W(B2) ⊆ W(B1). However, we know that if we extend the scope to allow p to be analytic, the conjecture is not true: Crouzeix has a whole family

• f counterexamples for n = 3.

Ice Cream Cone Fields of Values for f Closest to 1

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Optimizing over both p (deg ≤ n − 1) and A (order n) Optimizing over both p and A: Details Final Fields of Values for Lowest Computed f An Example: f(p, A) = 0.5000000002 A New Conjecture Ice Cream Cone Fields of Values for f Closest to 1 Nonsmooth Analysis

• f the Crouzeix

Ratio

39 / 50

• 1.5
• 1
• 0.5

0.5

• 1.5
• 1
• 0.5

0.5 1 1.5

n=3

• 4
• 2

2

• 4
• 3
• 2
• 1

1 2 3 4

n=4

• 8
• 6
• 4
• 2
• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

n=5

• 15
• 10
• 5

5

• 10
• 5

5 10

n=6

2 4

• 4
• 3
• 2
• 1

1 2 3 4

n=7

• 6
• 4
• 2

2

• 5
• 4
• 3
• 2
• 1

1 2 3 4 5

n=8

Perhaps the only stationary values of f are 0.5 and 1

Nonsmooth Analysis of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

40 / 50

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}.

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero.

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

F.H. Clarke, 1973 (he used the name “generalized gradient”).

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

F.H. Clarke, 1973 (he used the name “generalized gradient”). If h is continuously differentiable at ¯ x, then ∂h(¯ x) = {∇h(¯ x)}.

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

F.H. Clarke, 1973 (he used the name “generalized gradient”). If h is continuously differentiable at ¯ x, then ∂h(¯ x) = {∇h(¯ x)}. If h is convex, ∂h is the subdifferential of convex analysis.

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

F.H. Clarke, 1973 (he used the name “generalized gradient”). If h is continuously differentiable at ¯ x, then ∂h(¯ x) = {∇h(¯ x)}. If h is convex, ∂h is the subdifferential of convex analysis. We say ¯ x is Clarke stationary for h if 0 ∈ ∂h(¯ x) (a nonsmooth stationary point if ∈ ∂h(¯ x) contains more than one vector)

The Clarke Subdifferential

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

41 / 50

Assume h : Rn → R is locally Lipschitz, and let D = {x ∈ Rn : h is differentiable at x}. Rademacher’s Theorem: Rn\D has measure zero. The Clarke subdifferential, or set of subgradients, of h at ¯ x is ∂h(¯ x) = conv

• lim

x→¯ x,x∈D ∇h(x)

• .

F.H. Clarke, 1973 (he used the name “generalized gradient”). If h is continuously differentiable at ¯ x, then ∂h(¯ x) = {∇h(¯ x)}. If h is convex, ∂h is the subdifferential of convex analysis. We say ¯ x is Clarke stationary for h if 0 ∈ ∂h(¯ x) (a nonsmooth stationary point if ∈ ∂h(¯ x) contains more than one vector) Clarke stationarity is a necessary condition for local or global

• ptimality.

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator.

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator. Otherwise, need to take the convex hull of these gradients over all maximizing θ to get the subgradients of the numerator.

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator. Otherwise, need to take the convex hull of these gradients over all maximizing θ to get the subgradients of the numerator. For the denominator, combine:

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator. Otherwise, need to take the convex hull of these gradients over all maximizing θ to get the subgradients of the numerator. For the denominator, combine:

■

the gradient or subgradients of the 2-norm (maximum singular value) of a matrix (involves the singular vectors)

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator. Otherwise, need to take the convex hull of these gradients over all maximizing θ to get the subgradients of the numerator. For the denominator, combine:

■

the gradient or subgradients of the 2-norm (maximum singular value) of a matrix (involves the singular vectors)

■

the gradient of the matrix polynomial p(A) w.r.t. A (involves differentiating Ak w.r.t. A, resulting in Kronecker products).

The Gradient or Subgradients of the Crouzeix Ratio

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

42 / 50

For the numerator, we need the variational properties of max

θ∈[0,2π] |p(zθ)|

where zθ = v∗

θAvθ.

■

the gradient of p(zθ) w.r.t. the coefficients of p

■

the gradient of p(zθ) w.r.t. zθ

■

the gradient of zθ(A) = v∗

θAvθ w.r.t. A

If the max of |p(zθ)| is attained by a unique point ˆ θ, then all these are evaluated at ˆ θ and combined with the gradient of | · | to obtain the gradient of the numerator. Otherwise, need to take the convex hull of these gradients over all maximizing θ to get the subgradients of the numerator. For the denominator, combine:

■

the gradient or subgradients of the 2-norm (maximum singular value) of a matrix (involves the singular vectors)

■

the gradient of the matrix polynomial p(A) w.r.t. A (involves differentiating Ak w.r.t. A, resulting in Kronecker products).

Finally, use the quotient rule.

Regularity

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

43 / 50

A directionally differentiable, locally Lipschitz function h is regular (in the sense of Clarke, 1975) near a point x when its directional derivative x → h′(x; d) is upper semicontinuous there for every fixed direction d.

Regularity

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

43 / 50

A directionally differentiable, locally Lipschitz function h is regular (in the sense of Clarke, 1975) near a point x when its directional derivative x → h′(x; d) is upper semicontinuous there for every fixed direction d. In this case 0 ∈ ∂h(x) is equivalent to the first-order optimality condition h′(x, d) ≥ 0 for all directions d.

Regularity

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

43 / 50

A directionally differentiable, locally Lipschitz function h is regular (in the sense of Clarke, 1975) near a point x when its directional derivative x → h′(x; d) is upper semicontinuous there for every fixed direction d. In this case 0 ∈ ∂h(x) is equivalent to the first-order optimality condition h′(x, d) ≥ 0 for all directions d.

■

All convex functions are regular

Regularity

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

43 / 50

A directionally differentiable, locally Lipschitz function h is regular (in the sense of Clarke, 1975) near a point x when its directional derivative x → h′(x; d) is upper semicontinuous there for every fixed direction d. In this case 0 ∈ ∂h(x) is equivalent to the first-order optimality condition h′(x, d) ≥ 0 for all directions d.

■

All convex functions are regular

■

All continuously differentiable functions are regular

Regularity

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

43 / 50

A directionally differentiable, locally Lipschitz function h is regular (in the sense of Clarke, 1975) near a point x when its directional derivative x → h′(x; d) is upper semicontinuous there for every fixed direction d. In this case 0 ∈ ∂h(x) is equivalent to the first-order optimality condition h′(x, d) ≥ 0 for all directions d.

■

All convex functions are regular

■

All continuously differentiable functions are regular

■

Nonsmooth concave functions, e.g. h(x) = −|x|, are not regular.

Simplest Case where Crouzeix Ratio is Nonsmooth

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

44 / 50

Optimize over complex monic linear polynomials p(z) ≡ c + z and complex matrices with order n = 2. Let f(p, A) ≡ f(c, A), where now f : C × C2×2 → R.

Simplest Case where Crouzeix Ratio is Nonsmooth

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

44 / 50

Optimize over complex monic linear polynomials p(z) ≡ c + z and complex matrices with order n = 2. Let f(p, A) ≡ f(c, A), where now f : C × C2×2 → R. Let ˆ c = 0 (ˆ p(z) = z) and ˆ A = 2

• , so W( ˆ

A) = D, the unit disk, and hence |p(z)| is maximized everywhere on the unit circle, with f nonsmooth at (ˆ c, ˆ A) and f(ˆ c, ˆ A) = 1/2.

Simplest Case where Crouzeix Ratio is Nonsmooth

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

44 / 50

Optimize over complex monic linear polynomials p(z) ≡ c + z and complex matrices with order n = 2. Let f(p, A) ≡ f(c, A), where now f : C × C2×2 → R. Let ˆ c = 0 (ˆ p(z) = z) and ˆ A = 2

• , so W( ˆ

A) = D, the unit disk, and hence |p(z)| is maximized everywhere on the unit circle, with f nonsmooth at (ˆ c, ˆ A) and f(ˆ c, ˆ A) = 1/2. Theorem 3. The Crouzeix ratio f is regular at (ˆ c, ˆ A), with ∂f(ˆ c, ˆ A) = convθ∈[0,2π) 1 2e−iθ, 1 4 e−iθ e−2iθ e−iθ

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

45 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A)

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

45 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) Proof: the vectors inside the convex hull defined by θ = 0, 2π/3 and 4π/3 sum to zero.

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

45 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) Proof: the vectors inside the convex hull defined by θ = 0, 2π/3 and 4π/3 sum to zero. Actually, we knew this must be true as Crouzeix’s conjecture is known to hold for n = 2, and hence (ˆ c, ˆ A) is a global minimizer

• f f(·, ·), but we can extend the result to larger values of m, n,

for which we don’t know whether the conjecture holds.

The General Case

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

46 / 50

Optimize over complex polynomials p(z) ≡ c0 + · · · + cmzm and complex matrices with order n. Let f(p, A) ≡ f(c, A), where f : Cm+1 × Cn×n → R.

The General Case

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

46 / 50

Optimize over complex polynomials p(z) ≡ c0 + · · · + cmzm and complex matrices with order n. Let f(p, A) ≡ f(c, A), where f : Cm+1 × Cn×n → R. Let ˆ c = [0, 0, . . . , 1], corresponding to the polynomial zm, and ˆ A equal the C-matrix of order n = m + 1 so W( ˆ A) = D, the unit disk, and hence f(ˆ c, ˆ A) = 1/2.

The General Case

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

46 / 50

Optimize over complex polynomials p(z) ≡ c0 + · · · + cmzm and complex matrices with order n. Let f(p, A) ≡ f(c, A), where f : Cm+1 × Cn×n → R. Let ˆ c = [0, 0, . . . , 1], corresponding to the polynomial zm, and ˆ A equal the C-matrix of order n = m + 1 so W( ˆ A) = D, the unit disk, and hence f(ˆ c, ˆ A) = 1/2. Theorem 4. The Crouzeix ratio on (c, A) ∈ Cm+1 × Cn×n is regular at (ˆ c, ˆ A) with ∂f(ˆ c, ˆ A) = convθ∈[0,2π)

• yθ, Yθ
• where

yθ = 1 2 zm, zm−1, . . . , z, 0T and Yθ n × n matrix ˜ Yθ = 1 4          z √ 2z−1 √ 2z−2 · · · √ 2z3−n z2−n √ 2z2 2z 2z−1 · · · 2z4−n √ 2z3−n . . . . . . √ 2zn−2 2zn−3 2zn−4 2zn−5 · · · √ 2z √ 2zn−1 2zn−2 2zn−3 2zn−4 · · · 2z zn √ 2zn−1 √ 2zn−2 √ 2zn−3 · · · √ 2z2 z          with z = e−iθ.

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

47 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) so, for any n, the pair (ˆ c, ˆ A) is a nonsmooth stationary point of f.

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

47 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) so, for any n, the pair (ˆ c, ˆ A) is a nonsmooth stationary point of f.

• Proof. The convex combination

1 n + 1

n

• k=0
• y2kπ/(n+1), Y2kπ/(n+1)
• is zero.

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

47 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) so, for any n, the pair (ˆ c, ˆ A) is a nonsmooth stationary point of f.

• Proof. The convex combination

1 n + 1

n

• k=0
• y2kπ/(n+1), Y2kπ/(n+1)
• is zero.

This is a necessary condition for (ˆ c, ˆ A) to be a local (or global) minimizer of f on Rm+1 × Rn×n. This is a new result for n > 2.

(ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio The Clarke Subdifferential The Gradient or Subgradients of the Crouzeix Ratio Regularity Simplest Case where Crouzeix Ratio is Nonsmooth (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·) The General Case (ˆ c, ˆ A) is a Nonsmooth Stationary Point of f(·, ·)

47 / 50

Corollary. 0 ∈ ∂f(ˆ c, ˆ A) so, for any n, the pair (ˆ c, ˆ A) is a nonsmooth stationary point of f.

• Proof. The convex combination

1 n + 1

n

• k=0
• y2kπ/(n+1), Y2kπ/(n+1)
• is zero.

This is a necessary condition for (ˆ c, ˆ A) to be a local (or global) minimizer of f on Rm+1 × Rn×n. This is a new result for n > 2. And by regularity, it implies that the directional derivative f ′(·, d) ≥ 0 for all directions d.

Concluding Remarks

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

48 / 50

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Both Chebfun and BFGS perform remarkably reliably despite nonsmoothness that can occur either in the boundary of the field

• f values (w.r.t. the complex plane) or in the Crouzeix ratio

function (w.r.t the polynomial-matrix space).

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Both Chebfun and BFGS perform remarkably reliably despite nonsmoothness that can occur either in the boundary of the field

• f values (w.r.t. the complex plane) or in the Crouzeix ratio

function (w.r.t the polynomial-matrix space). Optimizing over p and A, BFGS essentially always converged either to nonsmooth stationary values of f associated with the C matrix (with field of values a disk), or smooth stationary values with “ice cream cone” fields of values.

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Both Chebfun and BFGS perform remarkably reliably despite nonsmoothness that can occur either in the boundary of the field

• f values (w.r.t. the complex plane) or in the Crouzeix ratio

function (w.r.t the polynomial-matrix space). Optimizing over p and A, BFGS essentially always converged either to nonsmooth stationary values of f associated with the C matrix (with field of values a disk), or smooth stationary values with “ice cream cone” fields of values. Using nonsmooth variational analysis, we proved Clarke stationarity of the Crouzeix ratio, with value 0.5, at pairs (˜ p, ˜ A), where ˜ p is the monomial zm and ˜ A is a C-matrix of order m + 1, a necessary condition for local or global optimality.

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Both Chebfun and BFGS perform remarkably reliably despite nonsmoothness that can occur either in the boundary of the field

• f values (w.r.t. the complex plane) or in the Crouzeix ratio

function (w.r.t the polynomial-matrix space). Optimizing over p and A, BFGS essentially always converged either to nonsmooth stationary values of f associated with the C matrix (with field of values a disk), or smooth stationary values with “ice cream cone” fields of values. Using nonsmooth variational analysis, we proved Clarke stationarity of the Crouzeix ratio, with value 0.5, at pairs (˜ p, ˜ A), where ˜ p is the monomial zm and ˜ A is a C-matrix of order m + 1, a necessary condition for local or global optimality. We also proved that given any other polynomial, there is a sequence of matrices of any given order for which the Crouzeix ratio 0.5 is approximated arbitrarily closely — but not attained.

Summary

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

49 / 50

Both Chebfun and BFGS perform remarkably reliably despite nonsmoothness that can occur either in the boundary of the field

• f values (w.r.t. the complex plane) or in the Crouzeix ratio

function (w.r.t the polynomial-matrix space). Optimizing over p and A, BFGS essentially always converged either to nonsmooth stationary values of f associated with the C matrix (with field of values a disk), or smooth stationary values with “ice cream cone” fields of values. Using nonsmooth variational analysis, we proved Clarke stationarity of the Crouzeix ratio, with value 0.5, at pairs (˜ p, ˜ A), where ˜ p is the monomial zm and ˜ A is a C-matrix of order m + 1, a necessary condition for local or global optimality. We also proved that given any other polynomial, there is a sequence of matrices of any given order for which the Crouzeix ratio 0.5 is approximated arbitrarily closely — but not attained. The results strongly support Crouzeix’s conjecture: the globally minimal value of the Crouzeix ratio f(p, A) is 0.5.

Our Papers

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

50 / 50

• A. Greenbaum and M.L. Overton

Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization Submitted to Linear Alg. Appl., October 2016

Our Papers

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

50 / 50

• A. Greenbaum and M.L. Overton

Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization Submitted to Linear Alg. Appl., October 2016

• A. Greenbaum, A.S. Lewis and M.L. Overton

Variational Analysis of the Crouzeix Ratio

• Math. Programming, 2016

Our Papers

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

50 / 50

• A. Greenbaum and M.L. Overton

Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization Submitted to Linear Alg. Appl., October 2016

• A. Greenbaum, A.S. Lewis and M.L. Overton

Variational Analysis of the Crouzeix Ratio

• Math. Programming, 2016

Both available at www.cs.nyu.edu/overton

Our Papers

Crouzeix’s Conjecture Nonsmooth Optimization of the Crouzeix Ratio f Fix p, Optimize over A Fix A, Optimize

• ver p

Optimizing over p and A Nonsmooth Analysis

• f the Crouzeix

Ratio Concluding Remarks Summary Our Papers

50 / 50

• A. Greenbaum and M.L. Overton

Investigation of Crouzeix’s Conjecture via Nonsmooth Optimization Submitted to Linear Alg. Appl., October 2016

• A. Greenbaum, A.S. Lewis and M.L. Overton

Variational Analysis of the Crouzeix Ratio

• Math. Programming, 2016

Both available at www.cs.nyu.edu/overton ¡ Muchas gracias por vuestra atencion !