A Verification Based Method to Generate Cutting Planes for IPs

Transcription

1 A Verification Based Method to Generate Cutting Planes for IPs Santanu S. Dey Sebastian Pokutta Georgia Institute of Technology, USA. Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. SIAM Conference on Optimization, 2011

2 Outline Cutting Plane Operators Generating Cutting Plane Differently: Design and Verify Verification Closure and Basic Properties Ranks using Verification Cuts Upper Bound on Ranks using Verification Cuts Lower Bound on Ranks using Verification Cuts

3 1 Cutting Plane Operators

4 A basic question in integer programming Question: Given a closed convex set P R n, obtain Convex hull of (P Z n )=: P I

5 A basic question in integer programming Question: Given a closed convex set P R n, obtain Convex hull of (P Z n )=: P I 1. This is a difficult task.

6 A basic question in integer programming Question: Given a closed convex set P R n, obtain Convex hull of (P Z n )=: P I 1. This is a difficult task. 2. Typically we are happy to obtain relaxation of the convex hull of P Z n, i.e., T R n s.t. P T P I. (1)

7 A basic question in integer programming Question: Given a closed convex set P R n, obtain Convex hull of (P Z n )=: P I 1. This is a difficult task. 2. Typically we are happy to obtain relaxation of the convex hull of P Z n, i.e., T R n s.t. P T P I. (1) 3. Let M be an cutting plane operator applied to P to obtain a (closed) convex relaxation of (P Z n ), i.e. P M(P) P I. (2)

8 Cutting planes: the way we use these operators If a, x b is a valid inequality for M(P), then it is a valid inequality for (P Z n ). Cutting Plane M(P) P

9 Examples of some well-known cutting-plane operators In this presentation, P R n " is always a rational polytope. Some operators for general integer programs: 1. Gomory-Chvátal closure (GC) 2. Split disjunctive closure (SC) Some operators for 0 1 integer programs, i.e, P [0, 1] n (Lovász-Schrijver operators): 1. Lift-and-project operator (N 0 ) 2. N 3. N +

10 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P.

11 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n.

12 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n. 3. HOMOGENEITY: M(F P) = F M(P), for all faces F of [0, 1] n.

13 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n. 3. HOMOGENEITY: M(F P) = F M(P), for all faces F of [0, 1] n. 4. SUBSTITUTION INDEPENDENCE: Let ϕ F be the projection onto the face F of [0, 1] n. Then ϕ F (M(P F )) = M(ϕ F (P F )).

14 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n. 3. HOMOGENEITY: M(F P) = F M(P), for all faces F of [0, 1] n. 4. SUBSTITUTION INDEPENDENCE: Let ϕ F be the projection onto the face F of [0, 1] n. Then ϕ F (M(P F )) = M(ϕ F (P F )). 5. SINGLE COORDINATE ROUNDING: If x i ɛ < 1 (or x i ɛ > 0) is valid for P, then x i 0 (or x i 1) is valid for M(P).

15 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n. 3. HOMOGENEITY: M(F P) = F M(P), for all faces F of [0, 1] n. 4. SUBSTITUTION INDEPENDENCE: Let ϕ F be the projection onto the face F of [0, 1] n. Then ϕ F (M(P F )) = M(ϕ F (P F )). 5. SINGLE COORDINATE ROUNDING: If x i ɛ < 1 (or x i ɛ > 0) is valid for P, then x i 0 (or x i 1) is valid for M(P). 6. COMMUTING WITH COORDINATE FLIPS AND DUPLICATIONS: τ i (M(P)) = M(τ i (P)), where τ i is either one of the following two operations: (i) Coordinate flip: τ i : [0, 1] n [0, 1] n with (τ i (x)) i = (1 x i ) and (τ i (x)) j = x j for j [n] \ {i}; (ii) Coordinate Duplication: τ i : [0, 1] n [0, 1] n+1 with (τ i (x)) n+1 = x i and (τ i (x)) j = x j for j [n].

16 Admissible cutting-plane procedures for 0-1 polytopes All known reasonable cutting plane operators satisfy the following properties: Definition A cutting-plane procedure M defined for a rational polytope P := {x [0, 1] n Ax b} is admissible if the following holds: 1. VALIDITY: P I M(P) P. 2. INCLUSION PRESERVATION: If P Q, then M(P) M(Q) for all polytopes P, Q [0, 1] n. 3. HOMOGENEITY: M(F P) = F M(P), for all faces F of [0, 1] n. 4. SUBSTITUTION INDEPENDENCE: Let ϕ F be the projection onto the face F of [0, 1] n. Then ϕ F (M(P F )) = M(ϕ F (P F )). 5. SINGLE COORDINATE ROUNDING: If x i ɛ < 1 (or x i ɛ > 0) is valid for P, then x i 0 (or x i 1) is valid for M(P). 6. COMMUTING WITH COORDINATE FLIPS AND DUPLICATIONS: τ i (M(P)) = M(τ i (P)), where τ i is either one of the following two operations: (i) Coordinate flip: τ i : [0, 1] n [0, 1] n with (τ i (x)) i = (1 x i ) and (τ i (x)) j = x j for j [n] \ {i}; (ii) Coordinate Duplication: τ i : [0, 1] n [0, 1] n+1 with (τ i (x)) n+1 = x i and (τ i (x)) j = x j for j [n]. 7. SHORT VERIFICATION: There exists a polynomial p such that for any inequality c, x d that is valid for M(P) there is a set I [m] with I p(n) such that c, x d is valid for M({x R n a i, x b i, i I}). Definition is modified from [Pokutta and Schulz (2009)]

17 Admissible cutting-plane procedures for general polytopes Definition If M is defined for general rational polytopes P R n, then we say M is admissible if M satisfies (1.)-(7.) when restricted to polytopes contained in [0, 1] n and for general polytopes P R n, M satisfies VALIDITY, INCLUSION PRESERVATION, SHORT VERIFICATION and HOMOGENEITY is replaced by 8. STRONG HOMOGENEITY: If P F := {x R n a, x b} and F = {x R n a, x = b} where (a, b) Z n Z, then M(F P) = M(P) F. Definition If M satisfies all required properties for being admissible except SHORT VERIFICATION, then we say M is almost admissible. Definition is modified from [Pokutta and Schulz (2009)]

18 2 Generating Cutting Plane Differently: Design and Verify

19 The Computation Scheme Input : P := {x R n Ax b} Output : Black box operator M c, x d valid for M(P)

20 Computation vs. Verification Scheme Computation Scheme Input : P := {x R n Ax b} Output : Black box operator M c, x d valid for M(P)

21 Computation vs. Verification Scheme Computation Scheme Input : P := {x R n Ax b} Output : Black box operator M c, x d valid for M(P) Verification Scheme Input : P := {x R n Ax b} Unfaithful Oracle to generate cuts Inequality claimed to be valid : c, x d Input : P := {x R n Ax b} Use Black Box operator to verify validity A similar idea discussed in Cook, Coullard, Turán (1987). Input : to do not print Use Black Box operator to verify validity do not print If valid, then Output : c, x d

22 How to verify validity of cut using black box operator Assuming c Z n, d Z: c, x d is valid for P I ( P {x R n c, x d + 1} ) I =

23 How to verify validity of cut using black box operator Assuming c Z n, d Z: c, x d is valid for P I M ( P {x R n c, x d + 1} ) = So if M (P {x R n c, x d + 1}) =, then we declare c, x d is a valid inequality.

24 Question: How much do we gain (if at all?) from having to only verify that a given inequality is valid for P I, rather than actually computing it.

25 Why is verification scheme interesting Sufficient condition for c, x d to be valid for P I :

26 Why is verification scheme interesting Sufficient condition for c, x d to be valid for P I : (i) M(P) {x c, x d + 1} = (computation)

27 Why is verification scheme interesting Sufficient condition for c, x d to be valid for P I : (i) M(P) {x c, x d + 1} = (computation) (ii) M ( a b P {x c, x d + 1} a b ) = (verification)

28 Why is verification scheme interesting Sufficient condition for c, x d to be valid for P I : (i) M(P) {x c, x d + 1} = (computation) (ii) M ( a b P {x c, x d + 1} a b ) = (verification) The strength of the verification scheme lies in the following inclusion that is satisfied by every "reasonable" cutting plane operator. M ( P {x R n c, x d + 1} ) M(P) {x R n c, x d + 1}.

29 Why is verification scheme interesting Sufficient condition for c, x d to be valid for P I : (i) M(P) {x c, x d + 1} = (computation) (ii) M ( a b P {x c, x d + 1} a b ) = (verification) The strength of the verification scheme lies in the following inclusion that is satisfied by every "reasonable" cutting plane operator. M ( P {x R n c, x d + 1} ) M(P) {x R n c, x d + 1}. This inclusion can be strict! 1. That is, suppose there exists (c 0, d 0 ) Z n+1 such that M(P {x c 0, x d 0 + 1}) = and (3) M(P) {x c 0, x d 0 + 1} = (4) 2. Then c 0, x d 0 is verifiable but not computable.

30 3 Verification Closure and Basic Properties

31 Verification Closure: Definition and Basic Property Verification Closure Definition Let M be admissible. Then M(P) := { c, x d M(P { c, x d + 1}) = } (5) (c,d) Z n+1 is the verification scheme closure of M. We add all inequalities whose validity can be verified with application of M.

32 Verification Closure: Definition and Basic Property Verification Closure Definition Let M be admissible. Then M(P) := { c, x d M(P { c, x d + 1}) = } (5) (c,d) Z n+1 is the verification scheme closure of M. We add all inequalities whose validity can be verified with application of M. Theorem If M is admissible, then M is almost admissible.

33 Comparing closures Properties of general admissible M Theorem 1. M(P) M(P) for all rational polytopes P. There exists a rational polytope Q, such that M(Q) M(Q). 2. M(P) GC(P) N 0 (P) for all rational polytopes P. There exists a rational polytope Q, such that M(Q) GC(Q) N 0 (Q).

34 Comparing closures Properties of general admissible M Theorem 1. M(P) M(P) for all rational polytopes P. There exists a rational polytope Q, such that M(Q) M(Q). 2. M(P) GC(P) N 0 (P) for all rational polytopes P. There exists a rational polytope Q, such that M(Q) GC(Q) N 0 (Q). Comparing specific closures Theorem Let L and M be admissible cutting plane operators such that L(P) M(P) for all rational polytopes P. Then L(P) M(P) for all rational polytopes P. Moreover, 1. GC(P) SC(P) for all rational polytopes P. There exists a rational polytope Q, such that GC(Q) SC(Q). 2. There exist polytopes Q 1 and Q 2, such that N 0 (Q 1 ) GC(Q 1 ) and N 0 (Q 2 ) GC(Q 2 ). 3. There exist polytopes Q 1 and Q 2, such that N 0 (Q 1 ) SC(Q 1 ) and N 0 (Q 2 ) SC(Q 2 ). 4. There exists a rational polytope Q, such that N(Q) N 0 (Q).

35 Comparing closures N0 pn0 GC N M SC pn N+ pm pgc pn+ psc

36 4.1 Upper Bound on Ranks using Verification Cuts

37 GC in R 2 Theorem GC(P) = P I for all rational polytopes P R 2, that is rk M (P) = 1 for all rational polytopes P R 2.

38 "Mimicking" the N + operator Lemma Let M be admissible and P [0, 1] n. Further let (c, d) Z n+1 +. If c, x d is valid for P {x x i = 1} for every i [n] with c i > 0, then c, x d is valid for M(P).

39 "Mimicking" the N + operator Lemma Let M be admissible and P [0, 1] n. Further let (c, d) Z n+1 +. If c, x d is valid for P {x x i = 1} for every i [n] with c i > 0, then c, x d is valid for M(P). Stable Set Polytope Given a graph G := (V, E), let FSTAB(G) := {x [0, 1] n x u + x v 1 (u, v) E}. Theorem Clique Inequalities, odd hole inequalities, odd anti-hole inequalities, and odd wheel inequalities are valid for M(FSTAB(G)) with M being an admissible operator. Proof uses previous lemma and ideas from [Lovász Schrijver (1991)].

40 "Mimicking" the N + operator Lemma Let M be admissible and P [0, 1] n. Further let (c, d) Z n+1 +. If c, x d is valid for P {x x i = 1} for every i [n] with c i > 0, then c, x d is valid for M(P). Monotone Polytope We say P [0, 1] n is monotone (or of anti-blocking type) if y P whenever y x and x P. Theorem Let M be admissible and P [0, 1] n be a monotone polytope with max x PI ex = k. Then rk M (P) k + 1. Proof uses previous lemma and ideas from [Cook Dash (2001)].

41 4.2 Lower Bound on Ranks using Verification Cuts

42 Not the paragon of perfection: lower bound on rank of A n A n := x [0, 1]n x i + (1 x i ) 1 2 i I i I I [n]. Lemma Let M be admissible and let l N such that rk M (A n) n rk M (A n) n 1. 2l + 1 l. If n > 2l + 1, then

43 Not the paragon of perfection: lower bound on rank of A n A n := x [0, 1]n x i + (1 x i ) 1 2 i I i I I [n]. Lemma Let M be admissible and let l N such that rk M (A n) n Corollary rk M (A n) n 1. 2l + 1 l. If n > 2l + 1, then Let M {GC, N 0, N, N +, SC} and n N with n 4. Then rk M (A n) n 1. 3

44 Not the paragon of perfection: lower bound on rank of A n Corollary Let M {GC, N 0, N, N +, SC} and n N with n 4. Then rk M (A n) Traveling Salesman Polytope n 1. 3 Let G = (V, E) be the undirected complete graph on n vertices. The subtour elimination polytope H n [0, 1] E is the set x(δ({v})) = 2 x(e(w )) W 1 v V W V x e [0, 1] e E. Theorem Let M {GC, SC, N 0, N, N +}. For n N and H n as defined above we have n/8 1 rk M (H n) n Proof uses results from [Chvátal, Cook, Hartmann (1989)] [Cook Dash (2001)].