Wolfes Combinatorial Method is Exponential Jamie Haddock STOC June - - PowerPoint PPT Presentation

Wolfe’s Combinatorial Method is Exponential

Jamie Haddock STOC June 26, 2018

UC Davis/UCLA

Poster in Bradbury/Rose and Hershey/Crocker at 8 pm

joint with Jes´ us A. De Loera and Luis Rademacher https://arxiv.org/abs/1710.02608

1

Minimum Norm Point in Polytope

We are interested in solving the problem (MNP(P)): min

x∈P x2

where P is a polytope, and determining the minimum dimension face, F, which achieves distance x2.

2

Minimum Norm Point in Polytope

We are interested in solving the problem (MNP(P)): min

x∈P x2

p1 p2 p3 x p5 p4 P O

2

Minimum Norm Point in Polytope

We are interested in solving the problem (MNP(P)): min

x∈P x2

p1 p2 p3 x p5 p4 P O ⊲ can be solved in polynomial time via interior-point methods

2

Minimum Norm Point in Polytope

We are interested in solving the problem (MNP(P)): min

x∈P x2

p1 p2 p3 x p5 p4 P O ⊲ can be solved in polynomial time via interior-point methods ⊲ no strongly-polynomial time algorithm known (even for simplex MNP)

2

Applications

• nearest point problem for transportation polytopes

3

Applications

• nearest point problem for transportation polytopes
• colorful linear programming

3

Applications

• nearest point problem for transportation polytopes
• colorful linear programming
• submodular function minimization

3

Applications

• nearest point problem for transportation polytopes
• colorful linear programming
• submodular function minimization

⊲ subroutine in Fujishige-Wolfe method

3

Applications

• nearest point problem for transportation polytopes
• colorful linear programming
• submodular function minimization

⊲ subroutine in Fujishige-Wolfe method ⊲ machine learning - vision, large-scale learning

3

Applications

• nearest point problem for transportation polytopes
• colorful linear programming
• submodular function minimization

⊲ subroutine in Fujishige-Wolfe method ⊲ machine learning - vision, large-scale learning

• linear programming

3

Our Results I

Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time.

4

Our Results I

Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time.

• Step 1: LP reduces to “membership in V -polytope.”

4

Our Results I

Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time.

• Step 1: LP reduces to “membership in V -polytope.”
• Step 2: “Membership in V -polytope” reduces to “distance to

V -simplex.”

4

Our Results I

Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time.

• Step 1: LP reduces to “membership in V -polytope.”
• Step 2: “Membership in V -polytope” reduces to “distance to

V -simplex.” If a strongly-polynomial method for projection onto a polytope exists then this gives a strongly-polynomial method for LP.

4

Our Results I

Theorem (De Loera, H., Rademacher ’17) Linear programming reduces to distance to a V -simplex in strongly-polynomial time.

• Step 1: LP reduces to “membership in V -polytope.”
• Step 2: “Membership in V -polytope” reduces to “distance to

V -simplex.” If a strongly-polynomial method for projection onto a polytope exists then this gives a strongly-polynomial method for LP. It was previously known that linear programming reduces to MNP on a polytope in weakly-polynomial time [Fujishige, Hayashi, Isotani ’06].

4

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

p1 p2 p3 x p5 p4 P O {y : xTy = x2

2} 5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)).

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O q1 q2 q3 O

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O q1 q2 q3 O

N N

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O q1 q2 q3 O

N N

q2 q1 O

N

X

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O q1 q2 q3 O

N N

Note: Singletons are corrals.

5

Background

Lemma (Wolfe ’74) Let P = conv(p1, p2, ..., pm). Then x ∈ P is MNP(P) if and only if xTpj ≥ x2

2 for all j = 1, 2, ..., m.

Def: An affinely independent set of input points Q = {q1, q2, ..., qk} is a corral if MNP(aff(Q)) ∈ relint(conv(Q)). q1 q2 O q1 q2 q3 O

N N

Note: Singletons are corrals. Note: There is a corral in P whose convex hull contains MNP(P).

5

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

• maintains set C defining the current corral and the MNP in C, x

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

• maintains set C defining the current corral and the MNP in C, x

Sketch: ⊲ Start with any point in P as C and x.

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

• maintains set C defining the current corral and the MNP in C, x

Sketch: ⊲ Start with any point in P as C and x. ⊲ If C not the optimal corral (checked via optimality criterion):

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

• maintains set C defining the current corral and the MNP in C, x

Sketch: ⊲ Start with any point in P as C and x. ⊲ If C not the optimal corral (checked via optimality criterion):

• Find improving point p (given via optimality criterion) to add to C.

6

Idea of Wolfe’s Method

• combinatorial method for computing MNP on a

vertex-representation polytope

• searches through sequence of corrals whose MNPs have strictly

decreasing norm until finding optimal

• maintains set C defining the current corral and the MNP in C, x

Sketch: ⊲ Start with any point in P as C and x. ⊲ If C not the optimal corral (checked via optimality criterion):

• Find improving point p (given via optimality criterion) to add to C.
• Let x “follow gravity” within conv(C) while dimension of current

face decreases to a face which is a corral and update C.

6

Related Methods

⊲ related to von Neumann’s algorithm for linear programming ⊲ related to Frank-Wolfe method for convex programming ⊲ related to Gilbert’s procedure for quadratic programming

7

Related Methods

⊲ related to von Neumann’s algorithm for linear programming ⊲ related to Frank-Wolfe method for convex programming ⊲ related to Gilbert’s procedure for quadratic programming ⊲ generalized by Hanson-Lawson procedure for non-negative least-squares

7

Our Results II

Theorem (De Loera, H., Rademacher ’17) There exists a set of points P(d) ⊂ Rd for d = 2k − 1 where Wolfe’s method with the minnorm insertion rule visits a sequence of corrals C(d)

• f length 5 · 2k−1 − 4.

8

Our Results II

Theorem (De Loera, H., Rademacher ’17) There exists a set of points P(d) ⊂ Rd for d = 2k − 1 where Wolfe’s method with the minnorm insertion rule visits a sequence of corrals C(d)

• f length 5 · 2k−1 − 4.

Note: P(d) is of size 2d − 1.

8

Our Results II

Theorem (De Loera, H., Rademacher ’17) There exists a set of points P(d) ⊂ Rd for d = 2k − 1 where Wolfe’s method with the minnorm insertion rule visits a sequence of corrals C(d)

• f length 5 · 2k−1 − 4.

Note: P(d) is of size 2d − 1. Note: binary encoding lengths of coordinates of points of P(d) grow polynomially in d.

8

Previous Results

• # iterations ≤ d+1

i=1 i

m

i

• with any insertion rules (Wolfe ’74)

9

Previous Results

• # iterations ≤ d+1

i=1 i

m

i

• with any insertion rules (Wolfe ’74)
• convergence rates for linopt insertion rule

(Chakrabarty, Jain, Kothari ’14) (Lacoste-Julien, Jaggi ’15)

9

Previous Results

• # iterations ≤ d+1

i=1 i

m

i

• with any insertion rules (Wolfe ’74)
• convergence rates for linopt insertion rule

(Chakrabarty, Jain, Kothari ’14) (Lacoste-Julien, Jaggi ’15)

9

Previous Results

• # iterations ≤ d+1

i=1 i

m

i

• with any insertion rules (Wolfe ’74)
• convergence rates for linopt insertion rule

(Chakrabarty, Jain, Kothari ’14) (Lacoste-Julien, Jaggi ’15) ⊲ pseudo-polynomial complexity

9

An Inefficient Example

⊲ a < p < q < r < s

10

An Inefficient Example

⊲ a < p < q < r < s

10

An Inefficient Example

⊲ a < p < q < r < s

10

An Inefficient Example

Sets C: a ap apq pq pqr qr qrs rs rsa ⊲ a < p < q < r < s

10

Exponential Lower Bound

⊲ replace a and x1 by a subspace and a set of points in it, constructed recursively

11

Exponential Lower Bound

⊲ replace a and x1 by a subspace and a set of points in it, constructed recursively ⊲ P(d) = P(d − 2) ∪ {pd, qd, rd, sd}

11

Exponential Lower Bound

⊲ replace a and x1 by a subspace and a set of points in it, constructed recursively ⊲ P(d) = P(d − 2) ∪ {pd, qd, rd, sd}

11

Exponential Lower Bound

⊲ replace a and x1 by a subspace and a set of points in it, constructed recursively ⊲ P(d) = P(d − 2) ∪ {pd, qd, rd, sd} Sets C: C(d − 2) O(d − 2)p pq qr rs rsC(d − 2)

11

Future Directions

12

Future Directions

• 1. Find an exponential example for Wolfe’s method with linopt

insertion rule.

12

Future Directions

• 1. Find an exponential example for Wolfe’s method with linopt

insertion rule.

• 2. Search for types of polytopes where Wolfe’s method is polynomial

(e.g. polytopes arising in submodular function minimizations).

12

Future Directions

• 1. Find an exponential example for Wolfe’s method with linopt

insertion rule.

• 2. Search for types of polytopes where Wolfe’s method is polynomial

(e.g. polytopes arising in submodular function minimizations).

• 3. Give an average (or smoothed) analysis of Wolfe’s method.

12

Thanks for attending!

Questions?

[1]

• I. B´

ar´ any and S. Onn. Colourful linear programming and its relatives. Mathematics of Operations Research, 22(3):550–567, 1997. [2]

• D. Chakrabarty, P. Jain, and P. Kothari.

Provable submodular minimization using Wolfe’s algorithm. In Proc. Advances Neural Info. Proc. Systems (NIPS), pages 802–809, 2014. [3]

• G. B. Dantzig.

An ǫ-precise feasible solution to a linear program with a convexity constraint in 1/ǫ2 iterations independent of problem size. Technical report, Stanford University, 1992. [4]

• J. A. De Loera, J. Haddock, and L. Rademacher.

The minimum Euclidean-norm point on a convex polytope: Wolfe’s combinatorial algorithm is exponential. 2017. [5]

• M. Frank and P. Wolfe.

An algorithm for quadratic programming. Naval Research Logistics (NRL), 3(1-2):95–110, 1956. [6]

• S. Fujishige, T. Hayashi, and S. Isotani.

The minimum-norm-point algorithm applied to submodular function minimization and linear programming. Citeseer, 2006. [7]

• E. G. Gilbert.

An iterative procedure for computing the minimum of a quadratic form on a convex set. SIAM J. Control, 4:61–80, 1966. [8]

• S. Lacoste-Julien and M. Jaggi.

On the global linear convergence of Frank-Wolfe

• ptimization variants.

In Proc. Advances Neural Info. Proc. Systems (NIPS), pages 496–504, 2015. [9]

• C. L. Lawson and R. J. Hanson.

Solving least squares problems, volume 15 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. Revised reprint of the 1974 original. [10]

• P. Wolfe.

Finding the nearest point in a polytope.

• Math. Programming, 11(2):128–149, 1976.

13

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P C = {p1}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P C = {p1}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < ||x||2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P C = {p1}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P y C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 = x p3 p2 O P y C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x = y C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x = y C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < ||x||2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x = y C = {p1, p2}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x = y C = {p1, p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O = y P x C = {p1, p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O = y P x C = {p1, p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

||z − y||2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) z p1 p3 p2 O = y P x C = {p1, p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) z p1 p3 p2 O = y P x C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) x = z p1 p3 p2 O = y P C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) x = z p1 p3 p2 O P y C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) x = z p1 p3 p2 O P y C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x = y C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x C = {p2, p3}

14

Sketch of Method

x ∈ P = {p1, p2, ..., pm} C = {x} while x is not MNP(P) pj ∈ {p ∈ P : xTp < x2

2}

C = C ∪ {pj} y = MNP(aff(C)) while y ∈ relint(conv(C)) z = argmin

z∈conv(C)∩xy

z − y2 C = C − {pi} where pi, z are on different faces of conv(C) x = z y = MNP(aff(C)) x = y return x p1 = (0, 2) p2 = (3, 0) p3 = (−2, 1) p1 p3 p2 O P x C = {p2, p3}

14