Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion Optimality Conditions for Edge-concave Quadratic Programs William Hager 1 James Hungerford 2 1 University of Florida 2 M.A.I.O.R. Srl, Italy MINO Initial Training Network January 7th, 2015 Aussois, France 1 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion Outline Introduction 1 Optimality conditions for ECQPs 2 Example: Vertex Separator Problem 3 Conclusion 4 2 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion Outline Introduction 1 Optimality conditions for ECQPs 2 Example: Vertex Separator Problem 3 Conclusion 4 3 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion min f ( x ) (1) x ∈ X s.t. 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 of X . 4 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion c T x min Ax ≤ b , x ∈ { 0 , 1 } n s.t. 5 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion c T x min Ax ≤ b , x ∈ { 0 , 1 } n s.t. c T x + γ x T ( 1 − x ) min (2) s.t. Ax ≤ b , 0 ≤ x ≤ 1 Theorem (Raghavachari, 1969) For sufficiently large γ , (2) has an optimal solution x ∗ ∈ { 0 , 1 } n . 5 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion Review of polyhedra P = { x ∈ R n : Ax ≤ b } A face of the polyhedron P is a non-empty set of the form H = { x ∈ P : ( Ax ) i = b i ∀ 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 ∈ R n is an edge direction if ∃ an edge H and some x ∈ H such that x + t d ∈ H for sufficiently small | t | . Definition (Edge description) A set D of edge directions is an edge description of P if each edge of P is parallel to some vector in D . If − d ∈ D whenever d ∈ D , then we say that D is reflective . 6 / 47
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 over P if for each x , y ∈ P such that y = x + t d for some d ∈ D and t ∈ R , we have f ( α x + (1 − α ) y ) ≥ αf ( x ) + (1 − α ) f ( y ) for every α ∈ [0 , 1] . Proposition If f ∈ C 2 ( P ) , then f is edge-concave over P if and only if d T ∇ 2 f ( x ) d ≤ 0 for each x ∈ P and for each d ∈ D . 7 / 47
Outline Introduction Optimality conditions for ECQPs Example: Vertex Separator Problem Conclusion min f ( x ) (3) x ∈ P s.t. 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 ) . 8 / 47
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 ℓ a , u a , the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓ a ≤ | A | ≤ u a , and the number of edges between A and B is minimized. �� �� ��� ��� �� �� ��� ��� �� �� ��� ��� A B 9 / 47
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 ℓ a , u a , the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓ a ≤ | A | ≤ u a , and the number of edges between A and B is minimized. ( 1 − x ) T ( A + I ) x min 0 ≤ x ≤ 1 and ℓ a ≤ 1 T x ≤ u a s.t. (Hager, Krylyuk, 1999) 10 / 47
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 ℓ a , u a , the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓ a ≤ | A | ≤ u a , and the number of edges between A and B is minimized. ( 1 − x ) T ( A + I ) x min 0 ≤ x ≤ 1 and ℓ a ≤ 1 T x ≤ u a s.t. D ⊆ ∪ n i,j =1 {± e i , e i − e j } 11 / 47
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 ℓ a , u a , the Edge Separator Problem is to partition V into two sets A, B ⊆ V , such that ℓ a ≤ | A | ≤ u a , and the number of edges between A and B is minimized. ( 1 − x ) T ( A + I ) x min 0 ≤ x ≤ 1 and ℓ a ≤ 1 T x ≤ u a s.t. e T i ( ∇ 2 f ) e i = − 1 ≤ 0 and ( e i − e j ) T ( ∇ 2 f )( e i − e j ) = (2 a ij − 2) ≤ 0 12 / 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 ℓ a , u a , ℓ b , u b , 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 , ℓ a ≤ | A | ≤ u a , ℓ b ≤ | B | ≤ u b , and | S | is minimized. A S B 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 ℓ a , u a , ℓ b , u b , 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 , ℓ a ≤ | A | ≤ u a , ℓ b ≤ | B | ≤ u b , and | S | is minimized. x T ( A + I ) y − 1 T ( x + y ) min s.t. 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , ℓ a ≤ 1 T x ≤ u a , and ℓ b ≤ 1 T y ≤ u b (Hager, Hungerford, 2014) 14 / 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 ℓ a , u a , ℓ b , u b , 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 , ℓ a ≤ | A | ≤ u a , ℓ b ≤ | B | ≤ u b , and | S | is minimized. x T ( A + I ) y − 1 T ( x + y ) min s.t. 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , ℓ a ≤ 1 T x ≤ u a , and ℓ b ≤ 1 T y ≤ u b D ⊆ ∪ n i,j =1 { [ ± e i , 0 ] , [ 0 , ± e i ] , [ e i − e j , 0 ] , [ 0 , e i − e j ] } 15 / 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 ℓ a , u a , ℓ b , u b , 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 , ℓ a ≤ | A | ≤ u a , ℓ b ≤ | B | ≤ u b , and | S | is minimized. x T ( A + I ) y − 1 T ( x + y ) min s.t. 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , ℓ a ≤ 1 T x ≤ u a , and ℓ b ≤ 1 T y ≤ u b d T ( ∇ 2 f ) d = 0 ≤ 0 for every d ∈ D 16 / 47
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. 17 / 47
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. 1 2 x T Ax − 1 T x min s.t. 0 ≤ x ≤ 1 (Harant, 2000) 18 / 47
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. 1 2 x T Ax − 1 T x min s.t. 0 ≤ x ≤ 1 D = ∪ n i =1 {± e i } 19 / 47
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. 1 2 x T Ax − 1 T x min s.t. 0 ≤ x ≤ 1 e T i ( ∇ 2 f ) e i = 0 ≤ 0 20 / 47
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