Optimality Conditions for Edge-concave Quadratic Programs William - - PowerPoint PPT Presentation

Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality Conditions for Edge-concave Quadratic Programs

William Hager1 James Hungerford2

1University of Florida 2M.A.I.O.R. Srl, Italy

MINO Initial Training Network

January 7th, 2015 Aussois, France

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Outline

1

Introduction

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Optimality conditions for ECQPs

3

Example: Vertex Separator Problem

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Conclusion

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Outline

1

Introduction

2

Optimality conditions for ECQPs

3

Example: Vertex Separator Problem

4

Conclusion

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

min f(x) (1) s.t. x ∈ X Theorem (Bauer, 1958) Suppose that f : X → R is concave and X is convex and compact. Then (1) has an optimal solution x∗ which lies at an extreme point

• f X.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

min cTx s.t. Ax ≤ b, x ∈ {0, 1}n

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

min cTx s.t. Ax ≤ b, x ∈ {0, 1}n min cTx + γxT(1 − x) (2) s.t. Ax ≤ b, 0 ≤ x ≤ 1 Theorem (Raghavachari, 1969) For sufficiently large γ, (2) has an optimal solution x∗ ∈ {0, 1}n.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Review of polyhedra

P = {x ∈ Rn : Ax ≤ b} A face of the polyhedron P is a non-empty set of the form H = {x ∈ P : (Ax)i = bi ∀ i ∈ I}, I ⊆ {1, 2, . . . , m}. If dim(H) = 0, then H is a vertex of P. If dim(H) = 1, then H is an edge of P. A vector d ∈ Rn is an edge direction if ∃ an edge H and some x ∈ H such that x + td ∈ H for sufficiently small |t|. Definition (Edge description) A set D of edge directions is an edge description of P if each edge

• f P is parallel to some vector in D. If −d ∈ D whenever d ∈ D,

then we say that D is reflective.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Definition (Edge-concavity) Let D be an edge description of P. A function f is edge-concave

• ver P if for each x, y ∈ P such that y = x + td for some d ∈ D

and t ∈ R, we have f(αx + (1 − α)y) ≥ αf(x) + (1 − α)f(y) for every α ∈ [0, 1]. Proposition If f ∈ C2(P), then f is edge-concave over P if and only if dT∇2f(x)d ≤ 0 for each x ∈ P and for each d ∈ D.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

min f(x) (3) s.t. x ∈ P Theorem (Tardella, 1990) Suppose f : P → R is edge-concave over P. If f attains its minimum on P, then (3) has an optimal solution x∗ which lies at a vertex of P. In this case, min f(x) = min f(x) s.t. x ∈ P s.t. x ∈ V(P).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Edge Separator Problem) Let G = (V, E) be a graph on vertex set V and edge set E. Given integers ℓa, ua, the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓa ≤ |A| ≤ ua, and the number of edges between A and B is minimized.

• A

B

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Edge Separator Problem) Let G = (V, E) be a graph on vertex set V and edge set E. Given integers ℓa, ua, the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓa ≤ |A| ≤ ua, and the number of edges between A and B is minimized. min (1 − x)T(A + I)x s.t. 0 ≤ x ≤ 1 and ℓa ≤ 1Tx ≤ ua (Hager, Krylyuk, 1999)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Edge Separator Problem) Let G = (V, E) be a graph on vertex set V and edge set E. Given integers ℓa, ua, the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓa ≤ |A| ≤ ua, and the number of edges between A and B is minimized. min (1 − x)T(A + I)x s.t. 0 ≤ x ≤ 1 and ℓa ≤ 1Tx ≤ ua D ⊆ ∪n

i,j=1{±ei, ei − ej}

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Edge Separator Problem) Let G = (V, E) be a graph on vertex set V and edge set E. Given integers ℓa, ua, the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓa ≤ |A| ≤ ua, and the number of edges between A and B is minimized. min (1 − x)T(A + I)x s.t. 0 ≤ x ≤ 1 and ℓa ≤ 1Tx ≤ ua eT

i (∇2f)ei = −1 ≤ 0 and

(ei − ej)T(∇2f)(ei − ej) = (2aij − 2) ≤ 0

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Vertex Separator Problem) Let G = (V, E) be a graph. Given integers ℓa, ua, ℓb, ub, the Vertex Separator Problem is to partition V into three sets A, B, S ⊆ V , such that there are no edges between A and B, ℓa ≤ |A| ≤ ua, ℓb ≤ |B| ≤ ub, and |S| is minimized.

S B A 13 / 47

Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Vertex Separator Problem) Let G = (V, E) be a graph. Given integers ℓa, ua, ℓb, ub, the Vertex Separator Problem is to partition V into three sets A, B, S ⊆ V , such that there are no edges between A and B, ℓa ≤ |A| ≤ ua, ℓb ≤ |B| ≤ ub, and |S| is minimized. min xT(A + I)y − 1T(x + y) s.t. 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ℓa ≤ 1Tx ≤ ua, and ℓb ≤ 1Ty ≤ ub (Hager, Hungerford, 2014)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Vertex Separator Problem) Let G = (V, E) be a graph. Given integers ℓa, ua, ℓb, ub, the Vertex Separator Problem is to partition V into three sets A, B, S ⊆ V , such that there are no edges between A and B, ℓa ≤ |A| ≤ ua, ℓb ≤ |B| ≤ ub, and |S| is minimized. min xT(A + I)y − 1T(x + y) s.t. 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ℓa ≤ 1Tx ≤ ua, and ℓb ≤ 1Ty ≤ ub D ⊆ ∪n

i,j=1{[±ei, 0], [0, ±ei], [ei − ej, 0], [0, ei − ej]}

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Vertex Separator Problem) Let G = (V, E) be a graph. Given integers ℓa, ua, ℓb, ub, the Vertex Separator Problem is to partition V into three sets A, B, S ⊆ V , such that there are no edges between A and B, ℓa ≤ |A| ≤ ua, ℓb ≤ |B| ≤ ub, and |S| is minimized. min xT(A + I)y − 1T(x + y) s.t. 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ℓa ≤ 1Tx ≤ ua, and ℓb ≤ 1Ty ≤ ub dT(∇2f)d = 0 ≤ 0 for every d ∈ D

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Maximum Stable Set Problem) Let G = (V, E) be a graph. The Maximum Stable Set Problem is to find the largest subset S ⊆ V such that S is stable; that is, no two vertices in S are adjacent.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Maximum Stable Set Problem) Let G = (V, E) be a graph. The Maximum Stable Set Problem is to find the largest subset S ⊆ V such that S is stable; that is, no two vertices in S are adjacent. min

1 2xTAx − 1Tx

s.t. 0 ≤ x ≤ 1 (Harant, 2000)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Maximum Stable Set Problem) Let G = (V, E) be a graph. The Maximum Stable Set Problem is to find the largest subset S ⊆ V such that S is stable; that is, no two vertices in S are adjacent. min

1 2xTAx − 1Tx

s.t. 0 ≤ x ≤ 1 D = ∪n

i=1{±ei}

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Examples of edge-concave formulations

Definition (Maximum Stable Set Problem) Let G = (V, E) be a graph. The Maximum Stable Set Problem is to find the largest subset S ⊆ V such that S is stable; that is, no two vertices in S are adjacent. min

1 2xTAx − 1Tx

s.t. 0 ≤ x ≤ 1 eT

i (∇2f)ei = 0 ≤ 0

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Local optimality in discrete and continuous problems

Optimality conditions in discrete problems Used in local search algorithms Based on an ad-hoc definition of “neighborhood” Checking local optimality is often in P Optimality conditions in continuous problems Used in iterative optimization algorithms Neighborhood is Bǫ(x) ∩ X Checking local optimality is NP-hard for general QPs (Pardalos, Schnitger, 1988)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Outline

1

Introduction

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Optimality conditions for ECQPs

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Example: Vertex Separator Problem

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Conclusion

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

General setting

min f(x) s.t. x ∈ P = {x ∈ Rn : Ax ≤ b} = ∅ A ∈ Rm×n with rank(A) = n (V(P) = ∅) f ∈ C2(P)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard first-order optimality condition

Given any x∗ ∈ P, the set of active constraints at x∗ is defined by A(x∗) = {i ∈ [1, m] : (Ax∗)i = bi}.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard first-order optimality condition

Given any x∗ ∈ P, the set of active constraints at x∗ is defined by A(x∗) = {i ∈ [1, m] : (Ax∗)i = bi}. The cone F(x∗) of first-order feasible directions at x∗ is F(x∗) = {d ∈ Rn : (Ad)i ≤ 0 for every i ∈ A(x∗)}.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard first-order optimality condition

Given any x∗ ∈ P, the set of active constraints at x∗ is defined by A(x∗) = {i ∈ [1, m] : (Ax∗)i = bi}. The cone F(x∗) of first-order feasible directions at x∗ is F(x∗) = {d ∈ Rn : (Ad)i ≤ 0 for every i ∈ A(x∗)}. Necessary condition: If x∗ ∈ P is a local minimizer, then ∇f(x∗)d ≥ 0 for every d ∈ F(x∗).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard second-order optimality conditions

The critical cone is defined by C(x∗) = {d ∈ F(x∗) : ∇f(x∗)d = 0} = {d ∈ F(x∗) : (Ad)i = 0 if λi > 0} = (a face of F(x∗)).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard second-order optimality conditions

The critical cone is defined by C(x∗) = {d ∈ F(x∗) : ∇f(x∗)d = 0} = {d ∈ F(x∗) : (Ad)i = 0 if λi > 0} = (a face of F(x∗)). Necessary Condition: If x∗ is a local minimizer, then dT∇2f(x∗)d ≥ 0 for every d ∈ C(x∗).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Standard second-order optimality conditions

The critical cone is defined by C(x∗) = {d ∈ F(x∗) : ∇f(x∗)d = 0} = {d ∈ F(x∗) : (Ad)i = 0 if λi > 0} = (a face of F(x∗)). Necessary Condition: If x∗ is a local minimizer, then dT∇2f(x∗)d ≥ 0 for every d ∈ C(x∗). Sufficient Condition: A feasible point x∗ is a local minimizer if the first-order condition holds and dT∇2f(x∗)d > 0 for every d ∈ C(x∗), d = 0.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge descriptions of the faces of F(x∗)

cone(X) = k

• i=1

αixi : xi ∈ X and αi ≥ 0 for every i

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge descriptions of the faces of F(x∗)

cone(X) = k

• i=1

αixi : xi ∈ X and αi ≥ 0 for every i

• Lemma

Every face H of F(x∗) is generated by the edge-directions of P lying in H. More precisely, if D is a reflective edge description of P, then H = cone(H ∩ D).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge descriptions of the faces of F(x∗)

cone(X) = k

• i=1

αixi : xi ∈ X and αi ≥ 0 for every i

• Corollary

Let x∗ ∈ P. Then F(x∗) = cone(F(x∗) ∩ D). Furthermore, if x∗ satisfies the first-order optimality conditions, then C(x∗) = cone(C(x∗) ∩ D).

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of first-order condition

Proposition (Necessary local optimality condition) Let D be a reflective edge description of P. A point x∗ ∈ P satisfies the standard first-order optimality condition if and only if ∇f(x∗)d ≥ 0 for every d ∈ F(x∗) ∩ D. (4)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of first-order condition

Proposition (Necessary local optimality condition) Let D be a reflective edge description of P. A point x∗ ∈ P satisfies the standard first-order optimality condition if and only if ∇f(x∗)d ≥ 0 for every d ∈ F(x∗) ∩ D. (4) Proof. ∇f(x∗)d = ∇f(x∗) k

• i=1

αidi

• =

k

• i=1

αi∇f(x∗)di.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of second-order condition

Proposition (Sufficient local optimality condition) Let x∗ ∈ P satisfy the first-order optimality condition. Then x∗ is a local minimizer if the following holds: (d1)T∇2f(x∗)d2 > 0 for every d1, d2 ∈ C(x∗) ∩ D. (5) If f is quadratic, then the strict inequality in (5) can be replaced by a nonstrict inequality.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of second-order condition

Proposition (Sufficient local optimality condition) Let x∗ ∈ P satisfy the first-order optimality condition. Then x∗ is a local minimizer if the following holds: (d1)T∇2f(x∗)d2 > 0 for every d1, d2 ∈ C(x∗) ∩ D. (5) If f is quadratic, then the strict inequality in (5) can be replaced by a nonstrict inequality. Proof. dT∇2f(x∗)d = k

i,j=1 αiαj(di)T∇2f(x∗)dj.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of second-order condition

Proposition (Necessary local optimality condition) Suppose that f is edge-concave at x∗. If x∗ is a local minimizer, then (d1)T∇2f(x∗)d2 ≥ 0 for every d1, d2 ∈ C(x∗) ∩ D.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Edge-reduction of second-order condition

Proposition (Necessary local optimality condition) Suppose that f is edge-concave at x∗. If x∗ is a local minimizer, then (d1)T∇2f(x∗)d2 ≥ 0 for every d1, d2 ∈ C(x∗) ∩ D. Proof. [dT∇2f(x∗)d ≥ 0 ∀ d ∈ C(x∗)] ⇒ [dT∇2f(x∗)d = 0 ∀ d ∈ C(x∗) ∩ D]. So, for every d1, d2 ∈ C(x∗) ∩ D we have 0 ≤ (d1 + d2)T∇2f(x∗)(d1 + d2) = 2(d1)T∇2f(x∗)d2.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality conditions for edge-concave QPs

Corollary (Necessary and sufficient local optimality conditions) If f is quadratic and edge-concave, then a point x∗ ∈ P is a local minimizer if and only if: ∇f(x∗)d ≥ 0 for every d ∈ F(x∗) ∩ D. (d1)T(∇2f)d2 ≥ 0 for every d1, d2 ∈ C(x∗) ∩ D.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality conditions for edge-concave QPs

Corollary (Necessary and sufficient local optimality conditions) If f is quadratic and edge-concave, then a point x∗ ∈ P is a local minimizer if and only if: ∇f(x∗)d ≥ 0 for every d ∈ F(x∗) ∩ D. (d1)T(∇2f)d2 ≥ 0 for every d1, d2 ∈ C(x∗) ∩ D. Moreover, if x∗ satisfies the first-order condition and not the second-order condition, then there exist d1, d2 ∈ D such that d = d1 + d2 is a descent direction.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality conditions for edge-concave QPs

Corollary (Necessary and sufficient local optimality conditions) If f is quadratic and edge-concave, then a point x∗ ∈ P is a local minimizer if and only if: ∇f(x∗)d ≥ 0 for every d ∈ F(x∗) ∩ D. (d1)T(∇2f)d2 ≥ 0 for every d1, d2 ∈ C(x∗) ∩ D. Moreover, if x∗ satisfies the first-order condition and not the second-order condition, then there exist d1, d2 ∈ D such that d = d1 + d2 is a descent direction. In this case, local optimality can be checked in time proportional to |D|2 ≤ 4 m n − 1 2 .

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality conditions for edge-concave QPs

Corollary (Polynomial (in m) complexity) If n is held fixed and f is quadratic and edge-concave, then local

• ptimality of any feasible point in P can be checked in time that is

bounded by a polynomial in m.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Optimality conditions for edge-concave QPs

Corollary (Polynomial (in m) complexity) If n is held fixed and f is quadratic and edge-concave, then local

• ptimality of any feasible point in P can be checked in time that is

bounded by a polynomial in m. Corollary (Polynomial (in n) complexity) Suppose that P = {x ∈ Rn : Bx ≤ b and ℓ ≤ x ≤ u}, where B ∈ Rl×n and b ∈ Rl. If l is held fixed and f is quadratic and edge-concave, then local optimality of any feasible point in P can be checked in time that is bounded by a polynomial in n.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Outline

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Introduction

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Optimality conditions for ECQPs

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Example: Vertex Separator Problem

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Conclusion

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Definition (Vertex Separator Problem) Let G = (V, E). Given integers ℓa, ua, ℓb, ub, the Vertex Separator Problem is to partition V into three sets A, B, S ⊆ V , such that there are no edges between A and B, ℓa ≤ |A| ≤ ua, ℓb ≤ |B| ≤ ub, and |S| is minimized. min xT(A + I)y − 1T(x + y) s.t. 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ℓa ≤ 1Tx ≤ ua, and ℓb ≤ 1Ty ≤ ub Aij = 1 if (i, j) ∈ E and 0 otherwise.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

min xT(A + I)y − 1T(x + y) s.t. 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ℓa ≤ 1Tx ≤ ua, and ℓb ≤ 1Ty ≤ ub Assume (x, y) satisfies the first-order optimality condition. (x, y) is a local minimizer if and only if (d1)T∇2f(x, y)d2 ≥ 0 for every d1, d2 ∈ C(x, y) ∩ D. D ⊆ ∪n

i,j=1{[±ei, 0], [0, ±ei], [ei − ej, 0], [0, ei − ej]}

∇2f(x, y) =

• A + I

A + I

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Example d1 = [ei, 0], d2 = [0, ej − ek], i = k = j d1 ∈ C(x, y) ⇔ 1Tx < ua, xi < 1, µa

i = λa = 0

d2 ∈ C(x, y) ⇔ yi < 1, yj > 0, µb

i = µb j = 0, µa i = λa = 0

(d1)T∇2f(x, y)d2 = eT

i (A + I)(ej − ek) = aij − aik

(d1)T∇2f(x, y)d2 ≤ 0 ⇔ aij ≤ aik ⇔ [(i, j) ∈ E ⇒ (i, k) ∈ E]

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Theorem (Necessary and sufficient conditions) Suppose (x, y) satisfies the first order optimality condition. Then (x, y) is a local minimizer if and only if the following hold: (1) Suppose λa = 0.

• a. If 1Tx < ua, i ∈ ¯

A0, j ∈ ¯ B0, and k ∈ B0, then hij ≥ hik.

• b. If 1Tx > ℓa, i ∈ A0, j ∈ B0, and k ∈ ¯

B0, then hij ≥ hik.

(2) Suppose λb = 0.

• a. If 1Ty < ub, i ∈ ¯

B0, j ∈ ¯ A0, and k ∈ A0, then hij ≥ hik.

• b. If 1Ty > ℓb, i ∈ B0, j ∈ A0, and k ∈ ¯

A0, then hij ≥ hik.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Theorem (Necessary and sufficient conditions) Suppose (x, y) satisfies the first order optimality condition. Then (x, y) is a local minimizer if and only if the following hold: (3) Suppose λa = λb = 0.

• a. If 1Tx > ℓa and 1Ty < ub, then A0 ∩ ¯

B0 = ∅ and hij = 0 whenever i ∈ A0 and j ∈ ¯ B0.

• b. If 1Tx < ua and 1Ty > ℓb, then ¯

A0 ∩ B0 = ∅ and hij = 0 whenever i ∈ ¯ A0 and j ∈ B0.

(4) If i ∈ ¯ A0, j ∈ A0, k ∈ ¯ B0, and l ∈ B0, then hik + hjl ≥ hil + hjk.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Definition (separable swap) Let (A, S, B) be a feasible partition of V . A series of vertex swaps is separable if no vertex in A or any of its neighbors is moved to B, and no vertex in B or any of its neighbors is moved to A. Example

S B A

1 1 2 3 2 2

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Definition (separable swap) Let (A, S, B) be a feasible partition of V . A series of vertex swaps is separable if no vertex in A or any of its neighbors is moved to B, and no vertex in B or any of its neighbors is moved to A. Example

S B A

1 3 2 2 1 2

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Definition (separable swap) Let (A, S, B) be a feasible partition of V . A series of vertex swaps is separable if no vertex in A or any of its neighbors is moved to B, and no vertex in B or any of its neighbors is moved to A. Example of nonseparable swap

• S

B A

1 1 1 2 3 1

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Definition (separable swap) Let (A, S, B) be a feasible partition of V . A series of vertex swaps is separable if no vertex in A or any of its neighbors is moved to B, and no vertex in B or any of its neighbors is moved to A. Example of nonseparable swap

• S

B A

1 1 1 3 1 2

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Example: Vertex Separator Problem

Proposition Let (A, S, B) be a feasible partition of V and let x and y be incidence vectors for A and B. Suppose that (x, y) satisfies the first-order optimality condition. Then,

• 1. There is no separable swap which improves the partition.
• 2. If (x, y) does not satisfy the second-order condition, then

there is an “easily computed” nonseparable swap improving the partition.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Outline

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Introduction

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Optimality conditions for ECQPs

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Example: Vertex Separator Problem

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Conclusion

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Conclusion

Some discrete optimization problems admit formulations as continuous edge-concave quadratic programs. Checking local optimality in an edge-concave quadratic program is easy. The optimality conditions for an edge-concave formulation can often be interpreted in a discrete way.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Conclusion

Some discrete optimization problems admit formulations as continuous edge-concave quadratic programs. Checking local optimality in an edge-concave quadratic program is easy. The optimality conditions for an edge-concave formulation can often be interpreted in a discrete way.

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Conclusion

Some discrete optimization problems admit formulations as continuous edge-concave quadratic programs. Checking local optimality in an edge-concave quadratic program is easy. The optimality conditions for an edge-concave formulation can often be interpreted in a discrete way. Hager, Hungerford, Optimality conditions for maximizing a function over a polyhedron, Mathematical Programming, no. 145,

• pg. 179-198 (2014)

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Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion

Thank you for your attention

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