A notion of Total Dual Integrality for Convex, Semidefinite and Extended Formulations

Transcription

1 A notion of for Convex, Semidefinite and Extended Formulations Marcel de Carli Silva Levent Tunçel April 26, 2018

2 A vector in R n is integral if each of its components is an integer,

3 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI)

4 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all.

5 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all. In this case, the polyhedron P determined by Ax b is integral, i.e., each nonempty face of P has an integral vector;

6 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all. In this case, the polyhedron P determined by Ax b is integral, i.e., each nonempty face of P has an integral vector; thus, (under the assumption that both primal and dual are feasible)

7 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all. In this case, the polyhedron P determined by Ax b is integral, i.e., each nonempty face of P has an integral vector; thus, (under the assumption that both primal and dual are feasible) equality holds throughout in the chain from the board.

8 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all. In this case, the polyhedron P determined by Ax b is integral, i.e., each nonempty face of P has an integral vector; thus, (under the assumption that both primal and dual are feasible) equality holds throughout in the chain from the board. This was proved in seminal work of Edmonds and Giles [1977] as a consequence of the following fundamental result:

9 A vector in R n is integral if each of its components is an integer, and a rational system of linear inequalities Ax b is called totally dual integral (TDI) if, for every integral vector c Z n, the LP problem dual to sup { c, x : Ax b} has an integral optimal solution whenever it has an optimal solution at all. In this case, the polyhedron P determined by Ax b is integral, i.e., each nonempty face of P has an integral vector; thus, (under the assumption that both primal and dual are feasible) equality holds throughout in the chain from the board. This was proved in seminal work of Edmonds and Giles [1977] as a consequence of the following fundamental result: Theorem (Edmonds-Giles [1977]) If A Q m n and b Q m satisfy sup { c, x : Ax b} Z {± } for each c Z n, then the polyhedron {x R n : Ax b} is integral.

10 (Edmonds and Giles [1977]) If A Q m n and b Q m satisfy sup { c, x : Ax b} Z {± } for each c Z n, then the polyhedron {x R n : Ax b} is integral.

11 (Edmonds and Giles [1977]) If A Q m n and b Q m satisfy sup { c, x : Ax b} Z {± } for each c Z n, then the polyhedron {x R n : Ax b} is integral. Corollary (Hoffman [1974]) Let A Q m n and b Q m. If P := {x R n : Ax b} is bounded and max x P c, x Z for each c Z n, then P is integral.

12 To define the integrality constraint for the dual SDP, we shall consider the dual slack.

13 To define the integrality constraint for the dual SDP, we shall consider the dual slack. Definition Let S be feasible in the dual SDP. We say that S is integral if S is a sum N S = k=1 S k of rank-one matrices S 1,..., S N S n+1 + k [N], we have such that, for each (S k ) 00 = 1, (S k ) 0j + (S k ) jj = 0 j [n].

14 If C R n is a compact convex set, then C = {x R n : w, x δ (C w) w Z n }.

15 If C R n is a compact convex set, then C = {x R n : w, x δ (C w) w Z n }. Let C I := conv (C Z n ).

16 If C R n is a compact convex set, then C = {x R n : w, x δ (C w) w Z n }. Let C I := conv (C Z n ). Using a bit more from IP theory, we conclude the following generalization of Hoffman s Theorem.

17 If C R n is a compact convex set, then C = {x R n : w, x δ (C w) w Z n }. Let C I := conv (C Z n ). Using a bit more from IP theory, we conclude the following generalization of Hoffman s Theorem. Corollary If C R n is a nonempty compact convex set, then C = C I if and only if δ (C w) Z for every w Z n.

18 Let C R n be a nonempty compact convex set. Then, TFAE

19 Let C R n be a nonempty compact convex set. Then, TFAE (i) C = C I ;

20 Let C R n be a nonempty compact convex set. Then, TFAE (i) C = C I ; (ii) every nonempty face of C contains an integral point;

21 Let C R n be a nonempty compact convex set. Then, TFAE (i) C = C I ; (ii) every nonempty face of C contains an integral point; (iii) for every w R n, max{ w, x : x C} is attained by an integral vector;

22 Let C R n be a nonempty compact convex set. Then, TFAE (i) C = C I ; (ii) every nonempty face of C contains an integral point; (iii) for every w R n, max{ w, x : x C} is attained by an integral vector; (iv) every rational supporting hyperplane for C contains integral points;

23 Let C R n be a nonempty compact convex set. Then, TFAE (i) C = C I ; (ii) every nonempty face of C contains an integral point; (iii) for every w R n, max{ w, x : x C} is attained by an integral vector; (iv) every rational supporting hyperplane for C contains integral points; (v) x C such that for each w Z n, { } 1 w, x + inf η R ++ : w (C x)o η Z.

24 Definition Let L : R k S n+1 be a linear map. The system A( ˆX ) b, ˆX 0 is TDI through L

25 Definition Let L : R k S n+1 be a linear map. The system A( ˆX ) b, ˆX 0 is TDI through L if for every c Z k, the SDP dual to { } sup L(c), ˆX : A( ˆX ) b ˆX 0 has an integral optimal solution, whenever it has an optimal solution.

26 Definition Let L : R k S n+1 be a linear map. The system A( ˆX ) b, ˆX 0 is TDI through L if for every c Z k, the SDP dual to { } sup L(c), ˆX : A( ˆX ) b ˆX 0 has an integral optimal solution, whenever it has an optimal solution. Note, ˆX is of the form [ ] 1 x x X

27 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1.

28 Let A( ˆX ) b, ˆX 0 be { TDI through a linear map} L: R k S n+1. Set Ĉ := ˆX S n+1 + : A( ˆX ) b and C := L (Ĉ) R k.

29 Let A( ˆX ) b, ˆX 0 be { TDI through a linear map} L: R k S n+1. Set Ĉ := ˆX S n+1 + : A( ˆX ) b and C := L (Ĉ) R k. If b is integral, C is compact, and Ĉ has a positive definite matrix,

30 Let A( ˆX ) b, ˆX 0 be { TDI through a linear map} L: R k S n+1. Set Ĉ := ˆX S n+1 + : A( ˆX ) b and C := L (Ĉ) R k. If b is integral, C is compact, and Ĉ has a positive definite matrix, then C = C I.

31 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that b is integral.

32 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix

33 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix and that C := L (Ĉ) [0, 1] k is compact.

34 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix and that C := L (Ĉ) [0, 1] k is compact. If Ĉ is a rank-one embedding of C I via L,

35 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix and that C := L (Ĉ) [0, 1] k is compact. If Ĉ is a rank-one embedding of C I via L, then for every w Z k, equality holds throughout in the chain of inequalities on the board,

36 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix and that C := L (Ĉ) [0, 1] k is compact. If Ĉ is a rank-one embedding of C I via L, then for every w Z k, equality holds throughout in the chain of inequalities on the board, all optimum values are equal to max { w, x : x C I },

37 Let A( ˆX ) b, ˆX 0 be TDI through a linear map L: R k S n+1 such that { b is integral. Suppose } that Ĉ := ˆX S n+1 + : A( ˆX ) b has a positive definite matrix and that C := L (Ĉ) [0, 1] k is compact. If Ĉ is a rank-one embedding of C I via L, then for every w Z k, equality holds throughout in the chain of inequalities on the board, all optimum values are equal to max { w, x : x C I }, and all suprema and infima are attained.

38 Open Problems/research Directions Obtain a primal-dual symmetric integrality condition for SDPs that applies to arbitrary ILPs, not just binary ones.

39 Open Problems/research Directions Obtain a primal-dual symmetric integrality condition for SDPs that applies to arbitrary ILPs, not just binary ones. Given k 1 and the LS + operator of Lovász and Schrijver [1991](called N + in their paper), determine the class of graphs for which the kth iterate of the LS + operator applied to the system x 0, x i + x j 1 ij E yields a TDI system through the appropriate lifting, leading to a family of combinatorial min-max theorems involving maximum weight stable sets in such graphs.

40 Open Problems/research Directions Obtain a primal-dual symmetric integrality condition for SDPs that applies to arbitrary ILPs, not just binary ones. Given k 1 and the LS + operator of Lovász and Schrijver [1991](called N + in their paper), determine the class of graphs for which the kth iterate of the LS + operator applied to the system x 0, x i + x j 1 ij E yields a TDI system through the appropriate lifting, leading to a family of combinatorial min-max theorems involving maximum weight stable sets in such graphs. Obtain systematic, primal-dual symmetric conditions for exactness in SDP relaxations for continuous problems.

41 Every pointed closed convex set is the intersection of all closed rational halfspaces containing it.

42 Every pointed closed convex set is the intersection of all closed rational halfspaces containing it. This presentation was based on: M. K. de Carli Silva and L. T., A notion of total dual integrality for convex, semidefinite, and extended formulations, arxiv: M. K. de Carli Silva and L. T., Pointed closed convex sets are the intersection of all rational supporting closed halfspaces, arxiv: