Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs. Pierre Bonami CMU, USA. Gerard Cornuéjols CMU, USA and LIF Marseille, France

Transcription

1 Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs Pierre Bonami CMU, USA Gerard Cornuéjols CMU, USA and LIF Marseille, France Sanjeeb Dash IBM T.J. Watson, USA Matteo Fischetti University of Padova, Italy Andrea Lodi University of Bologna, Italy Aussois X, January 9, 2006 A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs

2 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

3 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} and two associated polyhedra: P := {x IR n + : Ax b} P I := conv{x Z n + : Ax b} = conv(p Zn ) A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

4 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} and two associated polyhedra: P := {x IR n + : Ax b} P I := conv{x Z n + : Ax b} = conv(p Zn ) A Chvátal-Gomory (CG) cut is a valid inequality for P I of the form: A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

5 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} and two associated polyhedra: P := {x IR n + : Ax b} P I := conv{x Z n + : Ax b} = conv(p Zn ) A Chvátal-Gomory (CG) cut is a valid inequality for P I of the form: where u R m + u T A x u T b is called the CG multiplier vector, and denotes lower integer part. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

6 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} and two associated polyhedra: P := {x IR n + : Ax b} P I := conv{x Z n + : Ax b} = conv(p Zn ) A Chvátal-Gomory (CG) cut is a valid inequality for P I of the form: where u R m + u T A x u T b is called the CG multiplier vector, and denotes lower integer part. The first Chvátal closure of P is defined as: P 1 := {x 0 : Ax b, u T A x u T b for all u IR m + } A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

7 Notation and background Consider an Integer Linear Program (ILP) of the form: min{c T x : Ax b, x 0 integer} and two associated polyhedra: P := {x IR n + : Ax b} P I := conv{x Z n + : Ax b} = conv(p Zn ) A Chvátal-Gomory (CG) cut is a valid inequality for P I of the form: where u R m + u T A x u T b is called the CG multiplier vector, and denotes lower integer part. The first Chvátal closure of P is defined as: P 1 := {x 0 : Ax b, u T A x u T b for all u IR m + } P 1 is indeed a polyhedron, i.e., a finite number of CG cuts suffice to define it. [Chvátal 1973] A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 1

8 Notation and background (cont.d) Clearly, P I P 1 P. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

9 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

10 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

11 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

12 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

13 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. Thus, the natural question is: What does it happen in the Mixed IP case? A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

14 Notation and background (cont.d) Clearly, P I P 1 P. Chvátal-Gomory separation problem (CG-SEP) is NP-hard. [Eisenbrand 1999] Recently Fischetti & Lodi have shown that: Optimizing over P 1 is possible in practice via an MIP model (MIPping). P 1 is a good (often excellent) approximation of P I in practice. CG cuts in the first closure have a nice numerical behavior and stability. Thus, the natural question is: What does it happen in the Mixed IP case? Of course, the natural answer would be using Gomory Mixed Integer cuts (GMI) (also known as MIR cuts and split cuts) but their separation is much more involved than CG separation: nobody knows a MIP model for GMI yet! A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 2

15 Projected Chvátal-Gomory cuts Our first order of business is to extend the classical definition of Chvátal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 3

16 Projected Chvátal-Gomory cuts Our first order of business is to extend the classical definition of Chvátal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. We then consider the MIP: min{c T x + f T y : Ax + Cy b, x 0, x integral, y 0} with the two associated polyhedra: P (x, y) := {(x, y) R n + Rr + : Ax + Cy b} P I (x, y) := conv({(x, y) P (x, y) : x integral}) A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 3

17 Projected Chvátal-Gomory cuts Our first order of business is to extend the classical definition of Chvátal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. We then consider the MIP: min{c T x + f T y : Ax + Cy b, x 0, x integral, y 0} with the two associated polyhedra: P (x, y) := {(x, y) R n + Rr + : Ax + Cy b} P I (x, y) := conv({(x, y) P (x, y) : x integral}) and we project P (x, y) onto the space of x variables as: P (x) := {x R n + : there exists y Rr + s.t. Ax + Cy b} A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 3

18 Projected Chvátal-Gomory cuts Our first order of business is to extend the classical definition of Chvátal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. We then consider the MIP: min{c T x + f T y : Ax + Cy b, x 0, x integral, y 0} with the two associated polyhedra: P (x, y) := {(x, y) R n + Rr + : Ax + Cy b} P I (x, y) := conv({(x, y) P (x, y) : x integral}) and we project P (x, y) onto the space of x variables as: P (x) := {x R n + : there exists y Rr + s.t. Ax + Cy b} = {x R n + : uk A u k b, k = 1,..., K} A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 3

19 Projected Chvátal-Gomory cuts Our first order of business is to extend the classical definition of Chvátal-Gomory cuts to the mixed integer case, in such a way that the resulting separation problem remains a clean MIP. We then consider the MIP: min{c T x + f T y : Ax + Cy b, x 0, x integral, y 0} with the two associated polyhedra: P (x, y) := {(x, y) R n + Rr + : Ax + Cy b} P I (x, y) := conv({(x, y) P (x, y) : x integral}) and we project P (x, y) onto the space of x variables as: P (x) := {x R n + : there exists y Rr + s.t. Ax + Cy b} = {x R n + : uk A u k b, k = 1,..., K} =: {x R n + : Āx b} where u 1,..., u K are the (finitely many) extreme rays of the projection cone {u R m + : ut C 0 T }. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 3

20 Projected Chvátal-Gomory cuts (cont.d) We then define a projected Chvátal-Gomory (pro-cg) cut as a CG cut derived from the system Āx b, x 0, i.e., an inequality of the form w T Ā x w T b for some w 0. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 4

21 Projected Chvátal-Gomory cuts (cont.d) We then define a projected Chvátal-Gomory (pro-cg) cut as a CG cut derived from the system Āx b, x 0, i.e., an inequality of the form w T Ā x w T b for some w 0. More directly any pro-cg can be defined as an inequality of the form: u T A x u T b for any u 0 such that u T C 0 T A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 4

22 Projected Chvátal-Gomory cuts (cont.d) We then define a projected Chvátal-Gomory (pro-cg) cut as a CG cut derived from the system Āx b, x 0, i.e., an inequality of the form w T Ā x w T b for some w 0. More directly any pro-cg can be defined as an inequality of the form: u T A x u T b for any u 0 such that u T C 0 T As such, its associated separation problem can be modeled as: max α T x α 0 (1) α j u T A j, for j = 1,..., n (2) 0 u T C j, for j = 1,..., r (3) α ɛ u T b (4) u i 0, for i = 1,..., m (5) α j integer, for j = 0,..., n. (6) A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 4

23 On the strength of pro-cg cuts Given the above definition of pro-cg cuts it is straightforward to extend the definition of Chvátal-Gomory closure for the MIP case. We will denote as P 1 (x, y) the intersection of P (x, y) with all pro-cg cuts viewed as inequalities α T x + 0 T y α 0 in R n R r. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 5

24 On the strength of pro-cg cuts Given the above definition of pro-cg cuts it is straightforward to extend the definition of Chvátal-Gomory closure for the MIP case. We will denote as P 1 (x, y) the intersection of P (x, y) with all pro-cg cuts viewed as inequalities α T x + 0 T y α 0 in R n R r. For any π Z n and π 0 Z, the disjunction π T x π 0 or π T x π is valid for MIP. In other words, P I (x, y) conv(π 0 Π 1 ) where Π 0 := P (x, y) {(x, y) : π T x π 0 } Π 1 := P (x, y) {(x, y) : π T x π 0 + 1} A valid inequality for conv(π 0 Π 1 ) is called a split cut. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 5

25 On the strength of pro-cg cuts Given the above definition of pro-cg cuts it is straightforward to extend the definition of Chvátal-Gomory closure for the MIP case. We will denote as P 1 (x, y) the intersection of P (x, y) with all pro-cg cuts viewed as inequalities α T x + 0 T y α 0 in R n R r. For any π Z n and π 0 Z, the disjunction π T x π 0 or π T x π is valid for MIP. In other words, P I (x, y) conv(π 0 Π 1 ) where Π 0 := P (x, y) {(x, y) : π T x π 0 } Π 1 := P (x, y) {(x, y) : π T x π 0 + 1} A valid inequality for conv(π 0 Π 1 ) is called a split cut. A somehow expected result is: Theorem 1. Let S(x, y) denote the intersection of P (x, y) with all the split cuts where one of the sets Π 0, Π 1 defined aboveis empty. Then P 1 (x, y) = S(x, y). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 5

26 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

27 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

28 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). Thus, if the objective is to maximize x, pro-cg cuts help, and P 1 (x) = P I (x). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

29 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). Thus, if the objective is to maximize x, pro-cg cuts help, and P 1 (x) = P I (x). On the other hand, if the objective is to maximize y, pro-cg cuts do not help. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

30 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). Thus, if the objective is to maximize x, pro-cg cuts help, and P 1 (x) = P I (x). On the other hand, if the objective is to maximize y, pro-cg cuts do not help. More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y) belongs to the first Chvátal closure P 1 (x). In this case, no pro-cg cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

31 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). Thus, if the objective is to maximize x, pro-cg cuts help, and P 1 (x) = P I (x). On the other hand, if the objective is to maximize y, pro-cg cuts do not help. More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y) belongs to the first Chvátal closure P 1 (x). In this case, no pro-cg cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation. On the other hand, pro-cg cuts are well suited to handle those MIPs where the continuous variables are only used to model some feasibility condition, possibly by using big-m coefficients, but are not present in the objective function. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

32 On the strength of pro-cg cuts (cont.d) Consider the following simple example in two variables x and y: P (x, y) := {x + y 3/2, y x, x, y 0} Observe that the pro-cg cut x 1 cuts off the vertex (3/2, 0), but there is no pro-cg cut which cuts off the non-integral vertex (3/4, 3/4). Thus, if the objective is to maximize x, pro-cg cuts help, and P 1 (x) = P I (x). On the other hand, if the objective is to maximize y, pro-cg cuts do not help. More generally, suppose that the projection of the optimum of the MIP relaxation P (x, y) belongs to the first Chvátal closure P 1 (x). In this case, no pro-cg cut can cut off that point, although there might possibly be a huge gap between the MIP and its relaxation. On the other hand, pro-cg cuts are well suited to handle those MIPs where the continuous variables are only used to model some feasibility condition, possibly by using big-m coefficients, but are not present in the objective function. More precisely, it is not difficult to prove that: Theorem 2. MIPs where the continuous variables do not appear in the objective function can be optimized to proven optimality by using only pro-cg cuts (in an iterative way of course). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 6

33 How tight is the pro-cg closure for MIPLIB instances? Instances from MIPLIB 3.0 and MIPLIB 2003, time limit of 20 minutes, (% gap closed) = opt value(p I ) opt value(p 1 ) opt value(p I ) opt value(p ). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 7

34 How tight is the pro-cg closure for MIPLIB instances? Instances from MIPLIB 3.0 and MIPLIB 2003, time limit of 20 minutes, (% gap closed) = opt value(p I ) opt value(p 1 ) opt value(p I ) opt value(p ). pro-cg CPU % gap instance n r r c # iter # cuts time closed bell3a bell egout fixnet khb ,326 1, noswot rentacar 55 9, set1ch vpm vpm , mas mas misc , mod ,862 7, modglob pk rgn A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 7

35 How tight is the pro-cg closure for MIPLIB instances? (cont.d) pro-cg CPU % gap instance n r r c # iter # cuts time closed 10teams 1, ,001 1, arki , blend ,032 1, dcmulti , fiber 1, ,556 1, flugpl , gen , gesa ,660 1, gesa2 o , gesa , gesa3 o , mkc 5, , misc , misc , pp08a , qiu , qnet1 1, , qnet1 o 1, ,340 1, rout ,715 1, swath 6, ,222 1, A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 8

36 Is this a good result? pro-cg cuts have the advantage of being separated solving a simple MIP model and they seem to be effective (as already shown for CG cuts). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 9

37 Is this a good result? pro-cg cuts have the advantage of being separated solving a simple MIP model and they seem to be effective (as already shown for CG cuts). In order to additionally test their behavior we tested their effect when used in conjunction with other split cuts which are easy to separate. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 9

38 Is this a good result? pro-cg cuts have the advantage of being separated solving a simple MIP model and they seem to be effective (as already shown for CG cuts). In order to additionally test their behavior we tested their effect when used in conjunction with other split cuts which are easy to separate. More precisely, we applied one round of GMI and one round of MIR separated from the tableau of the initial continuous relaxation and we optimized over the lift-and-project closure (Bonami & Minoux) before starting separating pro-cg cuts (only using the initial constraint set Ax b). A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 9

39 Is this a good result? pro-cg cuts have the advantage of being separated solving a simple MIP model and they seem to be effective (as already shown for CG cuts). In order to additionally test their behavior we tested their effect when used in conjunction with other split cuts which are easy to separate. More precisely, we applied one round of GMI and one round of MIR separated from the tableau of the initial continuous relaxation and we optimized over the lift-and-project closure (Bonami & Minoux) before starting separating pro-cg cuts (only using the initial constraint set Ax b). % gap closed GMI GMI +MIR +MIR +L&P +L&P instance +pro-cg bell3a egout fixnet set1ch flugpl gesa2 o In other words, pro-cg cuts seem to be diverse wrt other cuts, thus increasing the arsenal of a cutting plane algorithm. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 9

40 Is this a good result? (cont.d) A final experiment to assert pro-cg s effectiveness is a comparison with the split closure itself. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 10

41 Is this a good result? (cont.d) A final experiment to assert pro-cg s effectiveness is a comparison with the split closure itself. A possible way of attacking the problem is looking at the separation of MIR cuts (Oktay Günlük, Sanjeeb Dash & Andrea Lodi): min c T + y + â T x ˆd( d ā T x ) s.t. â + ā λ T A, ā i Z, â i (0, 1) c + λ T C, c + 0 d λ T b A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 10

42 Is this a good result? (cont.d) A final experiment to assert pro-cg s effectiveness is a comparison with the split closure itself. A possible way of attacking the problem is looking at the separation of MIR cuts (Oktay Günlük, Sanjeeb Dash & Andrea Lodi): min c T + y + â T x ˆd( d ā T x ) s.t. â + ā λ T A, ā i Z, â i (0, 1) c + λ T C, c + 0 d λ T b and approximate it: Approximate ˆd as ˆd = k i=1 ɛ iπ i where ɛ i = 1 2 i, π i {0, 1} For any violated MIR cut, if = d ā T x, then 0 < < 1 A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 10

43 An Exact MIR separation model Lemma 1. The multipliers λ i corresponding to an equation without continuous variables can be assumed to lie in (0, 1) in an optimal solution to MIR-sep. Lemma 2. The multipliers λ i can be assumed to lie in ( mδ, mδ) in an optimal solution to MIR-sep, where m is the number of rows in P, and δ is the maximum value of sub-determinants of [A, C]. Corollary 1. The MIR closure of P is a polyhedron. Corollary 2. ˆd can be assumed to have finite precision. Corollary 3. MIR-sep can be solved as an MIP. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 11

44 Is this a good result? (cont.d) Of course the model is larger and more difficult to solve (binary variables for the approximation of d), thus, although theoretically dominating, the tradeoff of using such a model must be analyzed. A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 12

45 Is this a good result? (cont.d) Of course the model is larger and more difficult to solve (binary variables for the approximation of d), thus, although theoretically dominating, the tradeoff of using such a model must be analyzed. % gap closed instance pro-cg MIR bell3a blend dcmulti egout fixnet flugpl khb pp08a misc set1ch rgn vpm A. Lodi, Projected Chvátal-Gomory cuts for Mixed Integer Linear Programs 12