Some representability and duality results for convex mixed-integer programs.

Transcription

1 Some representability and duality results for convex mixed-integer programs. Santanu S. Dey Joint work with Diego Morán and Juan Pablo Vielma December 17, 2012.

2 Introduction About Motivation Mixed integer linear program min c T x s.t. Ax b, x Z n 1 R n 2, A, b are rational. A number of structural results are known about Integer Linear Programs: 1. Representability results of integer hulls ( Fundamental Theorem of Integer programming") 2. Subaddtive dual 3. Cutting plane closure, ranks, lengths, etc

3 3 Introduction About Motivation Mixed integer convex program min c T x s.t. x B, x Z n 1 R n 2, where B R n 1+n 2 is a closed convex set. In this case: 1. Representability results? 2. Dual? 3. Cutting plane closures?

4 Introduction About Motivation Mixed integer linear program min c T x A, b are rational. Results hold usually because: s.t. Ax b, x Z n 1 R n 2, 1. Finite number of extreme points of the LP relaxation each of which is rational, existence of basic feasible solutions for LPs. 2. Strong duality results for the Linear Programming (LP) relaxation, 3. Existence of finite generator of the Hilbert basis of a rational polyhedral cone.

5 Introduction About Motivation Mixed integer convex program min c T x s.t. x B, x Z n 1 R n 2, where B R n 1+n 2 is a closed convex set. In this case: 1. Finite number of extreme points of the LP relaxation each of which is rational, existence of basic feasible solutions for LPs. Does not hold 2. Strong duality results for the Linear Programming (LP) relaxation. 3. Existence of finite generator of the Hilbert basis of a rational polyhedral cone. Not always relevant in the convex case 5

6 6 Introduction About About this talk In this talk: Polyhedrality of integer hulls. (Representability-type result) Duality for Conic Mixed-integer programs. Generalizing the results for integer linear program to integer convex program.

7 6 Introduction About About this talk In this talk: Polyhedrality of integer hulls. (Representability-type result) Duality for Conic Mixed-integer programs. Generalizing the results for integer linear program to integer convex program. But first: A simple, but interesting property of convex mixed-integer programs: Finiteness Property.

8 7 An interesting property 1 Finiteness Property.

9 8 An interesting property Finiteness property Fact inf x B ct x > inf x B (Z n 1 R n 2) c T x >.

10 8 An interesting property Finiteness property Fact inf x B ct x > inf x B (Z n 1 R n 2) c T x >. Question ( ) inf x B ct x > inf x B (Z n 1 R n 2 ) c T x >???

11 8 An interesting property Finiteness property Fact inf x B ct x > inf x B (Z n 1 R n 2) c T x >. Question ( ) inf x B ct x > inf x B (Z n 1 R n 2 ) c T x >??? We know: B is polyhedron with rational data ( ) is true.

12 9 An interesting property Example 1: polyderal set with irrational data B is a line with irrational slope c Figure: ( ) is not true.

13 10 An interesting property Example 2: Second order conic representable (SOCR) set with rational data We will show a convex set B R 3 such that: conv(b Z 3 ) is a polyhedron. B is conic representable with rational data. There exists A Q m n, b Q m such that { ( } B = x R 3 x : u, A b L, u) where L is direct product of Lorentz cones. However, B does not satisfies the finiteness property ( ).

14 11 An interesting property Example: (SOCP) (cont.) B = conv(b 1 B 2 B 3 )

15 12 An interesting property Main result 1.1 Main result: A sufficient condition for the finiteness property.

16 13 An interesting property Main result A sufficient condition (for the finiteness property) Let B R n 1+n 2 be a closed convex set. Proposition If int(b) (Z n 1 R n 2), then ( ) is true, that is inf x B ct x > inf c T x >. x B (Z n 1 R n 2 )

17 14 An interesting property Sketch of the proof 1.3 Sketch of the proof. (In the pure integer case (n 2 = 0).)

18 15 An interesting property Sketch of the proof The convex set B. B

19 15 An interesting property Sketch of the proof The convex set B. Integer program is finite (z > ). z* -c B

20 15 An interesting property Sketch of the proof The convex set B. Integer program is finite (z > ). Yellow set B + is lattice-free. z* B + -c B

21 15 An interesting property Sketch of the proof The convex set B. Integer program is finite (z > ). Yellow set B + is lattice-free. exists maximal lattice-free convex set Q containing B +. Q = Polytope + Linear subspace. z* B + B -c

22 16 An interesting property Sketch of the proof Assume there exists {x 1, x 2, x 3... } B +, with c T x n. z* x 3 x 2 x 1 B + -c B

23 16 An interesting property Sketch of the proof Assume there exists {x 1, x 2, x 3... } B +, with c T x n. B is contains integer point in the interior B Q. x 0 x N x 3 x 2 x 1 B

24 16 An interesting property Sketch of the proof Assume there exists {x 1, x 2, x 3... } B +, with c T x n. B is contains integer point in the interior B Q. B + Q x 0 x N x 3 x 2 x 1 B

25 16 An interesting property Sketch of the proof Assume there exists {x 1, x 2, x 3... } B +, with c T x n. B is contains integer point in the interior B Q. B + Q x 0 x N x 3 x 2 x 1 for large N we obtain Contradiction! B

26 17 Polyhedrality of integer hulls 2 Polyhedrality of integer hulls.

27 18 Polyhedrality of integer hulls Integer hulls Theorem (Meyer, 1974) Let K R n be a polyhedron with rational polyhedral recession cone, then conv(k Z n ) is a rational polyhedron. This motivates the following questions:

28 18 Polyhedrality of integer hulls Integer hulls Theorem (Meyer, 1974) Let K R n be a polyhedron with rational polyhedral recession cone, then conv(k Z n ) is a rational polyhedron. This motivates the following questions: 1. Let K be a closed convex set. When is conv(k Z n ) a polyhedron?

29 19 Polyhedrality of integer hulls Examples of conv(k Z n ) being polyhedral

30 20 Polyhedrality of integer hulls Ex 1: K is a bounded convex set The easy case Figure: Polytope Figure: Bounded convex set

31 Polyhedrality of integer hulls Ex 2: K is an hyperbola

32 22 Polyhedrality of integer hulls Examples of conv(k Z n ) not being polyhedral

33 23 Polyhedrality of integer hulls Ex 1: K is a nonrational polyhedral cone Figure: conv(k Z n ) is not even closed

34 24 Polyhedrality of integer hulls Ex 2: K is a nonrational polyhedron Figure: conv(k Z n ) is not a polyhedron

35 25 Polyhedrality of integer hulls Ex 3: K has a rational polyhedral recession cone Figure: conv(k Z n ) is not a polyhedron

36 26 Polyhedrality of integer hulls Some remarks Based on the pictures What conditions on K led to a polyhedral conv(k Z n )? 1. Recession cone of K plays a fundamental role. Rational polyhedra recession cone.

37 26 Polyhedrality of integer hulls Some remarks Based on the pictures What conditions on K led to a polyhedral conv(k Z n )? 1. Recession cone of K plays a fundamental role. Rational polyhedra recession cone. 2. K needs to be similar to a polyhedron. Ex: Parabola vs Hyperbola.

38 Polyhedrality of integer hulls Developing intuition about kind of unboundedness Consider z = sup{c t x : x K } We have z =. u rec.cone(k ) = {λd, λ 0}. c d.

39 Polyhedrality of integer hulls Developing intuition about kind of unboundedness Consider z = sup{c t x : x K } We have z =. u rec.cone(k ) = {λd, λ 0}. c d.

40 28 Polyhedrality of integer hulls Main result 2.1 Main result Sufficient and necessary conditions for polyhedrality.

41 29 Polyhedrality of integer hulls Main result Definition Thin convex sets We say that a closed convex set K R n is thin if min{c T x x K } = d rec.cone(k ) c T d < 0. Every polyhedron is a thin convex set. Hyperbola is a thin convex set.

42 30 Polyhedrality of integer hulls Main result Necessary and sufficient conditions For polyhedrality Let K R n be a closed convex set. Theorem 1. If K is thin and rec.cone(k ) is a rational polyhedral cone, then conv(k Z n ) is a polyhedron.

43 30 Polyhedrality of integer hulls Main result Necessary and sufficient conditions For polyhedrality Let K R n be a closed convex set. Theorem 1. If K is thin and rec.cone(k ) is a rational polyhedral cone, then conv(k Z n ) is a polyhedron. 2. If int(k ) Z n and conv(k Z n ) is a polyhedron, then K is thin and rec.cone(k ) is a rational polyhedral cone.

44 31 Polyhedrality of integer hulls Sketch of the proof 2.2 Sketch of the proof.

45 32 Polyhedrality of integer hulls Sketch of the proof Gordan-Dickson Lemma (GDL) Lemma Let m N and S Z m and assume that s 0 Z m such that for all s S s s 0. Then there exists T S, finite set, satisfying for all s S t T such that s t.

46 33 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 1. rec.cone(k ) = {d R n : Ad 0} for some A Z m n. rec.cone(conv(k Z n )) = {d R n : Ad 0}

47 33 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 1. rec.cone(k ) = {d R n : Ad 0} for some A Z m n. rec.cone(conv(k Z n )) = {d R n : Ad 0} 2. We want to show polyhedrality conv(k Z n ) = conv(y ) + rec.cone(k ), for a finite set Y K Z n.

48 33 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 1. rec.cone(k ) = {d R n : Ad 0} for some A Z m n. rec.cone(conv(k Z n )) = {d R n : Ad 0} 2. We want to show polyhedrality conv(k Z n ) = conv(y ) + rec.cone(k ), for a finite set Y K Z n. 3. It suffices to show that for a finite set Y K Z n. K Z n Y + rec.cone(k ),

49 34 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 4. Let S = {Ax : x K Z n } Z m.

50 34 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 4. Let S = {Ax : x K Z n } Z m. 5. K is thin and rec.cone(k ) = {d R n : Ad 0} s 0 Z m such that Ax s 0 x K Z n. s 0 Z m such that s s 0 s S.

51 34 Polyhedrality of integer hulls Sketch of the proof Sufficient condition Sketch of Proof 4. Let S = {Ax : x K Z n } Z m. 5. K is thin and rec.cone(k ) = {d R n : Ad 0} s 0 Z m such that Ax s 0 x K Z n. s 0 Z m such that s s 0 s S. 6. We can apply GDL to S to obtain a finite set T S. Y K Z n, finite set, such that T = AY. for all x K Z n y Y such that Ax Ay. K Z n Y + rec.cone(k ).

52 35 Polyhedrality of integer hulls Sketch of the proof Necessary condition Sketch of Proof 1. We know that conv(k Z n ) = {x R n : Ax b}, for some A Z m n, b Z m. 2. Since int(k ) Z n, by finiteness property, b Z m such that K {x R n : Ax b }. 3. We obtain that {x R n : Ax b} K {x R n : Ax b }

53 Polyhedrality of integer hulls Sketch of the proof Necessary condition Sketch of Proof 1. We know that conv(k Z n ) = {x R n : Ax b}, for some A Z m n, b Z m. 2. Since int(k ) Z n, by finiteness property, b Z m such that K {x R n : Ax b }. 3. We obtain that {x R n : Ax b} K {x R n : Ax b } rec.cone(k ) = {x Z n : Ax 0}, a rational polyhedral cone. K is a thin set. 35

54 36 Duality for conic mixed-integer programs 3 Duality for conic mixed-integer programs

55 37 Duality for conic mixed-integer programs Notation and basic ideas 3.1 Notation and basic ideas

56 38 Duality for conic mixed-integer programs Notation and basic ideas The primal problems Let A R m n, c R n, b R m and I {1,..., n}. (MILP): inf c T x (P MILP ) s.t. Ax b x i Z, i I. (u v u v R n +)

57 38 Duality for conic mixed-integer programs Notation and basic ideas The primal problems Let A R m n, c R n, b R m and I {1,..., n}. (MILP): inf c T x (P MILP ) s.t. Ax b x i Z, i I. (MICP): inf c T x (P) s.t. Ax K b x i Z, i I. (u v u v R n +) (u K v u v K )

58 39 Duality for conic mixed-integer programs Notation and basic ideas Some definitions Subadditivity, conic non-decreasing Let Ω R m, and g : Ω R { } be a function. Definition (Subadditive function) g is said to be subadditive if for all u, v Ω u + v Ω g(u + v) g(u) + g(v). Definition (Nondecreasing w.r.t K ) g is said to be nondecreasing w.r.t. K if for u, v Ω u K v g(u) g(v).

59 40 Duality for conic mixed-integer programs Notation and basic ideas The dual problem (D MILP ) The Subadditive dual for (P MILP ) sup g(b) ( s.t. g A i) ( = g A i) = c i, ( (D MILP ) ḡ A i) ( = ḡ A i) = c i, i I i / I g(0) = 0 g : R m R, subadditive, nondecreasing w.r.t. R n +. where A i is the ith column of A, and g(d) = lim sup δ 0 + g(δd) δ.

60 41 Duality for conic mixed-integer programs Notation and basic ideas The dual problem (D) The Subadditive dual for (P) sup g(b) ( s.t. g A i) ( = g A i) = c i, ( (D) ḡ A i) ( = ḡ A i) = c i, i I i / I g(0) = 0 g : R m R, subadditive, nondecreasing w.r.t. to K. Remark When K = R m + we retrieve the subadditive dual for (P MILP ).

61 Duality for conic mixed-integer programs Notation and basic ideas Properties of a nice dual Correct lower bounds: for all x R n feasible for (P) and for all g : R m R feasible for (D), we have g(b) c T x (Weak duality).

62 Duality for conic mixed-integer programs Notation and basic ideas Properties of a nice dual Correct lower bounds: for all x R n feasible for (P) and for all g : R m R feasible for (D), we have g(b) c T x (Weak duality). Similar behavior: (P) is feasible and bounded (D) is feasible and bounded

63 Duality for conic mixed-integer programs Notation and basic ideas Properties of a nice dual Correct lower bounds: for all x R n feasible for (P) and for all g : R m R feasible for (D), we have g(b) c T x (Weak duality). Similar behavior: (P) is feasible and bounded (D) is feasible and bounded Best possible lower bounds: (P) is feasible and bounded, then there exists g : R m R feasible for (D) such that g (b) = inf{c T x : Ax K b, x i Z, i I} (Strong duality).

64 43 Duality for conic mixed-integer programs MILP case Theorem Let A Q m n, b Q m (data is rational). Then (D MILP ) is a nice dual for (P MILP ).

65 43 Duality for conic mixed-integer programs MILP case Theorem Let A Q m n, b Q m (data is rational). Then (D MILP ) is a nice dual for (P MILP ).

66 Duality for conic mixed-integer programs MILP case Some applications (of subadditive duality for (P MILP )) Dual feasible functions give ALL valid linear inequalities for (P MILP ): g(a i )x i + g(a i )x i g(b), i I i C where g is a feasible dual function. Example Gomory mixed-integer cuts are given by dual feasible functions corresponding to 1-row Mixed integer linear programs.

67 45 Duality for conic mixed-integer programs Main result 3.2 Main result: Extension of the Subadditive duality theory for (MILP) to the case of (MICP).

68 46 Duality for conic mixed-integer programs Main result Strong duality for (MILP) The well-known result: Theorem (Strong duality for (MILP)) If A Q m n, b Q m (data is rational), then 1. (P MILP ) is feasible and bounded if and only if (D MILP ) is feasible and bounded. 2. If (P MILP ) is feasible and bounded, then there exists a function g feasible for (D MILP ) such that g (b) = inf{c T x : Ax b, x i Z, i I}.

69 Duality for conic mixed-integer programs Main result Strong duality for (MICP) The new result: Theorem (Strong duality for (MICP)) If there exists ˆx Z n 1 R n 2 s.t. Aˆx K b, then 1. (P) is feasible and bounded if and only if (D) is feasible and bounded. 2. If (P) is feasible and bounded, then there exists a function g feasible for (D) such that g (b) = inf{c T x : Ax K b, x i Z, i I}.

70 48 Duality for conic mixed-integer programs Sketch of the proof 2 Sketch of the proof.

71 49 Duality for conic mixed-integer programs Sketch of the proof Basic proof steps P R : continuous relaxation of the primal. D R : dual of P R.

72 49 Duality for conic mixed-integer programs Sketch of the proof Basic proof steps P R : continuous relaxation of the primal. D R : dual of P R. 1. (P) is feasible and bounded (D) is feasible and bounded. ( ): (P) is feasible and bounded (P R ) is feasible and bounded. (D R ) is feasible. (D) is feasible. Weak duality (D) bounded. ( ): basically same proof as (MILP) case.

73 50 Duality for conic mixed-integer programs Sketch of the proof Basic proof steps If (P MILP ) is feasible and bounded, then there exists a function g feasible for (D MILP ) such that g (b) = inf{c T x : Ax b, x i Z, i I}. 1. Properties of value function f Definition: f : Ω R { }, defined as Domain of f : f (u) = inf{c T x : Ax K u, x i Z, i I}. u Ω (P) with r.h.s. u is feasible. Property of f : In general, f satisfies all constraints of (D), except by Ω = R m.

74 51 Duality for conic mixed-integer programs Sketch of the proof Basic proof steps 2. If (P) is feasible and bounded, then there exists a function g feasible for (D) such that g (b) = z. If Ω = R m, we just can take g = f. Because f (b) = inf{c T x : Ax K b, x i Z, i I}.

75 51 Duality for conic mixed-integer programs Sketch of the proof Basic proof steps 2. If (P) is feasible and bounded, then there exists a function g feasible for (D) such that g (b) = z. If Ω = R m, we just can take g = f. Because f (b) = inf{c T x : Ax K b, x i Z, i I}. Otherwise, if Ω R m we extend f to a function g such that g is feasible for the dual (D). g (b) = f (b). This extension is very technical and uses the fact that there exists ˆx Z n 1 R n 2 s.t. Aˆx K b.

76 52 Duality for conic mixed-integer programs Sketch of the proof Final remarks The sufficient condition for strong duality is: In (MILP): rational data. In (MICP): int(b) (Z n 1 R n 2).

77 52 Duality for conic mixed-integer programs Sketch of the proof Final remarks The sufficient condition for strong duality is: In (MILP): rational data. In (MICP): int(b) (Z n 1 R n 2). The condition: ˆx Z n 1 R n 2 s.t. Aˆx K b is used in: Strong duality for Conic programming. Finiteness property. Extension of value function.

78 53 Thanks Acknowledgement Many thanks to Diego Morán to help prepare these slides.