Flyspeck Inequalities and Semidefinite Programming Victor Magron , - - PowerPoint PPT Presentation

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Flyspeck Inequalities and Semidefinite Programming Victor Magron , - - PowerPoint PPT Presentation

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Flyspeck Inequalities and Semidefinite Programming Victor Magron , RA Imperial College Memory Optimization and Co-Design Meeting 29 June 2015 Victor Magron Flyspeck Inequalities and Semidefinite Programming 1 / 18 Errors and Proofs

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SLIDE 1

Flyspeck Inequalities and Semidefinite Programming

Victor Magron, RA Imperial College Memory Optimization and Co-Design Meeting 29 June 2015

Victor Magron Flyspeck Inequalities and Semidefinite Programming 1 / 18

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Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

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Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

  • M. Lecat, Erreurs des Mathématiciens des origines à nos

jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, Sylvester, . . . )

Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

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SLIDE 4

Errors and Proofs

Mathematicians and Computer Scientists want to eliminate all the uncertainties on their results. Why?

  • M. Lecat, Erreurs des Mathématiciens des origines à nos

jours, 1935. ❀ 130 pages of errors! (Euler, Fermat, Sylvester, . . . ) Ariane 5 launch failure, Pentium FDIV bug

Victor Magron Flyspeck Inequalities and Semidefinite Programming 2 / 18

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SLIDE 5

Errors and Proofs

Possible workaround: proof assistants COQ (Coquand, Huet 1984) HOL-LIGHT (Harrison, Gordon 1980) Built in top of OCAML

Victor Magron Flyspeck Inequalities and Semidefinite Programming 3 / 18

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Complex Proofs

Complex mathematical proofs / mandatory computation

  • K. Appel and W. Haken , Every Planar Map is

Four-Colorable, 1989.

  • T. Hales, A Proof of the Kepler Conjecture, 1994.

Victor Magron Flyspeck Inequalities and Semidefinite Programming 4 / 18

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SLIDE 7

From Oranges Stack...

Kepler Conjecture (1611): The maximal density of sphere packings in 3D-space is

π √ 18

Face-centered cubic Packing Hexagonal Compact Packing

Victor Magron Flyspeck Inequalities and Semidefinite Programming 5 / 18

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...to Flyspeck Nonlinear Inequalities

The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: “[...] the mathematical community will have to get used to this state of affairs.” Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture

Victor Magron Flyspeck Inequalities and Semidefinite Programming 6 / 18

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...to Flyspeck Nonlinear Inequalities

The proof of T. Hales (1998) contains mathematical and computational parts Computation: check thousands of nonlinear inequalities Robert MacPherson, editor of The Annals of Mathematics: “[...] the mathematical community will have to get used to this state of affairs.” Flyspeck [Hales 06]: Formal Proof of Kepler Conjecture Project Completion on 10 August by the Flyspeck team!!

Victor Magron Flyspeck Inequalities and Semidefinite Programming 6 / 18

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...to Flyspeck Nonlinear Inequalities

Nonlinear inequalities: quantified reasoning with “∀” ∀x ∈ K, f (x) 0 NP-hard optimization problem

Victor Magron Flyspeck Inequalities and Semidefinite Programming 7 / 18

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A “Simple” Example

In the computational part: Multivariate Polynomials:

∆x := x1x4(−x1 + x2 + x3 − x4 + x5 + x6) + x2x5(x1 − x2 + x3 + x4 − x5 + x6) + x3x6(x1 + x2 − x3 + x4 + x5 − x6) − x2(x3x4 + x1x6) − x5(x1x3 + x4x6)

Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

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A “Simple” Example

In the computational part: Semialgebraic functions: composition of polynomials with | · |, √, +, −, ×, /, sup, inf, . . . p(x) := ∂4∆x q(x) := 4x1∆x r(x) := p(x)/

  • q(x)

l(x) := −π 2 + 1.6294 − 0.2213 (√x2 + √x3 + √x5 + √x6 − 8.0) + 0.913 (√x4 − 2.52) + 0.728 (√x1 − 2.0)

Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

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A “Simple” Example

In the computational part: Transcendental functions T : composition of semialgebraic functions with arctan, exp, sin, +, −, ×, . . .

Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

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A “Simple” Example

In the computational part: Feasible set K := [4, 6.3504]3 × [6.3504, 8] × [4, 6.3504]2 Lemma9922699028 from Flyspeck: ∀x ∈ K, arctan p(x)

  • q(x)
  • + l(x) 0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 8 / 18

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Existing Formal Frameworks

Formal proofs for Global Optimization: Bernstein polynomial methods [Zumkeller’s PhD 08] SMT methods [Gao et al. 12] Interval analysis and Sums of squares

Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

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Existing Formal Frameworks

Interval analysis Certified interval arithmetic in COQ [Melquiond 12] Taylor methods in HOL Light [Solovyev thesis 13]

Formal verification of floating-point operations

robust but subject to the Curse of Dimensionality

Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

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Existing Formal Frameworks

Lemma9922699028 from Flyspeck: ∀x ∈ K, arctan

  • ∂4∆x

√4x1∆x

  • + l(x) 0

Dependency issue using Interval Calculus:

One can bound ∂4∆x/√4x1∆x and l(x) separately Too coarse lower bound: −0.87 Subdivide K to prove the inequality

K = ⇒ K0 K1 K2 K3 K4

Victor Magron Flyspeck Inequalities and Semidefinite Programming 9 / 18

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Introduction Flyspeck Inequalities and Semidefinite Programming

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Semidefinite Programming

Linear Programming (LP): min

z

c

⊤z

s.t. A z d .

Linear cost c Linear inequalities “∑i Aij zj di”

Polyhedron

Victor Magron Flyspeck Inequalities and Semidefinite Programming 10 / 18

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Semidefinite Programming

Semidefinite Programming (SDP): min

z

c

⊤z

s.t.

∑

i

Fi zi F0 .

Linear cost c Symmetric matrices F0, Fi Linear matrix inequalities “F 0” (F has nonnegative eigenvalues)

Spectrahedron

Victor Magron Flyspeck Inequalities and Semidefinite Programming 11 / 18

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Semidefinite Programming

Semidefinite Programming (SDP): min

z

c

⊤z

s.t.

∑

i

Fi zi F0 , A z = d .

Linear cost c Symmetric matrices F0, Fi Linear matrix inequalities “F 0” (F has nonnegative eigenvalues)

Spectrahedron

Victor Magron Flyspeck Inequalities and Semidefinite Programming 12 / 18

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SDP for Polynomial Optimization

Prove polynomial inequalities with SDP: p(a, b) := a2 − 2ab + b2 0 . Find z s.t. p(a, b) =

  • a

b z1 z2 z2 z3

  • a

b

  • .

Find z s.t. a2 − 2ab + b2 = z1a2 + 2z2ab + z3b2 (A z = d)

z1 z2 z2 z3

  • =

1

  • F1

z1 + 1 1

  • F2

z2 + 1

  • F3

z3

  • F0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 13 / 18

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SDP for Polynomial Optimization

Choose a cost c e.g. (1, 0, 1) and solve: min

z

c

⊤z

s.t.

∑

i

Fi zi F0 , A z = d . Solution

z1 z2 z2 z3

  • =

1 −1 −1 1

  • (eigenvalues 0 and 1)

a2 − 2ab + b2 =

  • a

b 1 −1 −1 1

  • a

b

  • = (a − b)2 .

Solving SDP = ⇒ Finding SUMS OF SQUARES certificates

Victor Magron Flyspeck Inequalities and Semidefinite Programming 14 / 18

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Polynomial Optimization

Semidefinite Programming

  1 a b a 1 c b c 1   0

control, polynomial optim (Henrion, Lasserre, Parrilo) combinatorial optim. electrical engineering(Laurent, Steurers)

Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

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Polynomial Optimization

Semidefinite Programming

  1 a b a 1 c b c 1   0

control, polynomial optim (Henrion, Lasserre, Parrilo) combinatorial optim. electrical engineering(Laurent, Steurers)

Theoretical Approach INFINITE LP p∗ := infRn p(x) ? sup λ ⇐ with p − λ 0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

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Polynomial Optimization

Semidefinite Programming

  1 a b a 1 c b c 1   0

control, polynomial optim (Henrion, Lasserre, Parrilo) combinatorial optim. electrical engineering(Laurent, Steurers)

Practical Approach FINITE SDP p∗ := infRn p(x) ? sup λ ⇐ with p − λ = sums of squares

  • f fixed degree

Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

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Polynomial Optimization

Semidefinite Programming

  1 a b a 1 c b c 1   0

control, polynomial optim (Henrion, Lasserre, Parrilo) combinatorial optim. electrical engineering(Laurent, Steurers)

Practical Approach FINITE SDP p∗ := infRn p(x) ? sup λ ⇐ with p − λ = sums of squares

  • f fixed degree

SDP bounds Hierarchy ↑ p∗ degree d n variables

= ⇒ (n+2d

n ) variables SDP

Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

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Polynomial Optimization

Semidefinite Programming

  1 a b a 1 c b c 1   0

control, polynomial optim (Henrion, Lasserre, Parrilo) combinatorial optim. electrical engineering(Laurent, Steurers)

Practical Approach FINITE SDP p∗ := infRn p(x) ? sup λ ⇐ with p − λ = sums of squares

  • f fixed degree

SDP bounds Hierarchy ↑ p∗ degree d n variables

= ⇒ (n+2d

n ) variables SDP

Strengthening p − λ = sums of squares = ⇒ p λ

1 + x4

1 − 2x2 1x2 2 + x4 2 = 1 + (x2 1 − x2 2)2

Victor Magron Flyspeck Inequalities and Semidefinite Programming 15 / 18

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Non-polynomial Optimization

TAYLOR + INTERVALS : + scalable − coarse

K = ⇒ K0 K1 K2 K3 K4

Curse of dimensionality

Victor Magron Flyspeck Inequalities and Semidefinite Programming 16 / 18

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Non-polynomial Optimization

TAYLOR + INTERVALS : + scalable − coarse

K = ⇒ K0 K1 K2 K3 K4

Curse of dimensionality TAYLOR + SUMS OF SQUARES : − not scalable + precise high degree d n variables

= ⇒ (n+2d

n )

No free lunch

Victor Magron Flyspeck Inequalities and Semidefinite Programming 16 / 18

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Non-polynomial Optimization

TAYLOR + INTERVALS : + scalable − coarse

K = ⇒ K0 K1 K2 K3 K4

Curse of dimensionality TAYLOR + SUMS OF SQUARES : − not scalable + precise high degree d n variables

= ⇒ (n+2d

n )

No free lunch MAXPLUS + SUMS OF SQUARES : + scalable + precise Maxplus in control (Akian Gaubert)

  • Templates in static analysis (Manna)

Curse reduction Maxplus Approximations Approximate f : Rn → R with supremum of quadratic forms.

Victor Magron Flyspeck Inequalities and Semidefinite Programming 16 / 18

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Non-polynomial Optimization

MAXPLUS + SUMS OF SQUARES: + scalable + precise Function from “simple” inequality:

+ l(x) arctan r(x) Victor Magron Flyspeck Inequalities and Semidefinite Programming 17 / 18

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Non-polynomial Optimization

MAXPLUS + SUMS OF SQUARES: + scalable + precise Verification software NLCertify, 1st iteration:

+ l(x) arctan r(x) a y par−

a1

arctan m M a1

1 control point {a1} m1 = −4.7 × 10−3 < 0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 17 / 18

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Non-polynomial Optimization

MAXPLUS + SUMS OF SQUARES: + scalable + precise Verification software NLCertify, 2nd iteration:

+ l(x) arctan r(x) a y par−

a1

par−

a2

arctan m M a1 a2

2 control points {a1, a2} m2 = −6.1 × 10−5 < 0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 17 / 18

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Non-polynomial Optimization

MAXPLUS + SUMS OF SQUARES: + scalable + precise Verification software NLCertify, 3rd iteration:

+ l(x) arctan r(x) a y par−

a1

par−

a2

par−

a3

arctan m M a1 a2 a3

3 control points {a1, a2, a3} m3 = 4.1 × 10−6 > 0

Victor Magron Flyspeck Inequalities and Semidefinite Programming 17 / 18

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Non-polynomial Optimization

MAXPLUS + SUMS OF SQUARES: + scalable + precise Verification software NLCertify, 3rd iteration:

+ l(x) arctan r(x) a y par−

a1

par−

a2

par−

a3

arctan m M a1 a2 a3

3 control points {a1, a2, a3} m3 = 4.1 × 10−6 > 0

Theorem The algorithm converges to a global optimum and certifies inequalities. nHales : time ratio between formal and numerical certification (V.

Vœvodsky)

❀ nHales 10 (Maxplus + Sums of squares) ≪ 2000 (Taylor + Intervals)

Victor Magron Flyspeck Inequalities and Semidefinite Programming 17 / 18

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Contributions

CERTIFICATION MAXPLUS–SUMS OF SQUARES: NUMERIC

OR FORMAL

Journals Magron, Allamigeon, Gaubert & Werner, Journal Math. Prog. Ser. B 2014 Magron, Allamigeon, Gaubert & Werner, Journal of Formalized Reasoning 2015 Conferences Allamigeon, Gaubert, Magron & Werner, Calculemus Conference 2013 Allamigeon, Gaubert, Magron & Werner, European Control Conference 2013 Magron, ICMS Conference 2014 + software NLCertify A FORMAL PROOF OF KEPLER CONJECTURE Hales, Adams, Bauer, Dang, Harrison, Hoang, Kaliszyk, Magron, Mclaughlin, Nguyen, Nguyen, Nipkow, Obua, Pleso, Rute, Solovyev, Ta, Tran, Trieu, Urban, Vu & Zumkeller, Prepublication, submitted Sigma/Pi Journal 2015

Victor Magron Flyspeck Inequalities and Semidefinite Programming 18 / 18

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End

Thank you for your attention! cas.ee.ic.ac.uk/people/vmagron