Integer program reformulation for robust branch-and-cut-and-price

Transcription

1 Integer program reformulation for robust branch-and-cut-and-price Marcus Poggi de Aragão Informática PUC-Rio Eduardo Uchoa Engenharia de Produção Universidade Federal Fluminense

2 Outline of the talk Robust branch-and-cut-and-price Dantzig-Wolfe master X Explicit master Tackling symmetries in the original formulation Computational Results Capacitated Vehicle Routing Problem Capacitated Minimum Spanning Tree Problem Generalized Assignment Problem Perspectives

3 What is a robust branch-and-cut-and-price? A branch-and-bound algorithm where: each node solves a linear program with an exponential number of both variables and constraints, neither branching nor separation (cutting) ever change the structure of the pricing subproblem.

4 Robust branch-and-cut-and-price Since cut and column generation were established as two of the most important techniques in IP, one looks for ways of combining them. Big (partially open) question: How to perform cutting without changing the structure of the pricing subproblem?

5 Robust branch-and-cut-and-price First (non-robust) BCP: Nemhauser and Park (1991), edge-coloring problem. In a non-robust BCP, the pricing subproblem eventually needs to be solved by general IP techniques. This only works for small instances. An exception: Belov and Scheithauer ( ), on a number of generalized cutting stock problems where column generation gives very tight bounds. Mixed-integer Gomory cuts may be enough to close the gaps in the root node.

6 Robust branch-and-cut-and-price A number of researchers noted that cuts expressed in terms of variables from an original formulation do not disturb the pricing: Vanderbeck (1998), lot-sizing. Kim, Barnhart, Ware and Reinhardt (1999), a service network design problem. Kohl, Desrosiers, Madsen, Solomon and Soumis (1999), VRPTW. Felici, Gentile and Rinaldi (2000), a supply-chain problem. Barnhart, Hane and Vance (2001), integer multiflow.

7 Robust branch-and-cut-and-price This approach has a fundamental drawback as a general IP technique: on many important problems the original formulation sufers from variable symmetries. Cuts over those variables are ineffective. Implementing an efficient robust BCP raises many new practical questions. Up to now, robust BCP was only performed on problems where BP alone already performs well. We feel that the applicability of this technique goes far beyond that class of problems.

8 IP Reformulation Let (O) be the following IP with n variables: (O) min s. t cx Ax Dx = b d x N n { } P = x N Dx d n and suppose is a finite set with elements x 1, x 2,...x p.

9 IP Reformulation Let Q be a n X p matrix where each column corresponds to one element of P There is a one-to-one correspondence between elements of P and the solutions of: s. t x = Qλ 1λ = 1 λ { 0,1} p

10 IP Reformulation The traditional IP reformulation (Gilmore and Gomory, 1961) consists of replacing x by its equivalent expression in (O), a transformation similar to the Dantzig-Wolfe reformulation for LP: Z IP s. t = min ( cq) λ ( AQ) λ = b 1λ = 1 λ { 0,1} p

11 Dantzig-Wolfe Master ( DWM ) ZDWM = min ( cq) λ s. t ( AQ) λ = b 1λ = 1 λ 0 Solving the DWM is equivalent to solving Z DWM s. t Ax = b = min cx { n } x Conv x N Dx d

12 Dantzig-Wolfe Master Columns on DWM are generated by solving the following IP: V ( µ, ν ) = min( c µ A) x ν s. t Dx d x N n, where µ and ν are the associated dual variables.

13 Cutting on the Dantzig-Wolfe Master Let λ be a fractional solution of DWM. On the original formulation, this corresponds i to solution x = Qλ. A cut a x bi violated by x can be added to DWM in order to cut λ. min ( cq) λ s. t ( AQ) λ = b i ( a Q) λ 1λ = λ 0 i Incorporating the new row a to A and the new dual variable µ i to µ, the pricing subproblem remains unchanged. b i 1

14 Alternative IP Reformulation Rewrite (O) by adding variables x : (O') min cx s. t. x ' x = 0 Ax Dx ' x ' x = b d N N n n

15 Alternative IP Reformulation Replacing x by its equivalent expression, we obtain another reformulation: x Q s. t 1 1 0,1 p min cx s. t. Qλ x Ax = = 0 b 1λ λ = 1 0 x N n

16 Explicit Master (EM ) min cx s. t. Qλ x = 0 Ax = b 1λ λ x = 1 0 0

17 Explicit Master Columns on EM are generated by solving the following IP: V ( π, ν ) = min( π x) ν s. t Dx d x N n, Dual variables µ associated to Ax = b do not appear in the subproblem.

18 Cutting on the Explicit Master (EM ) Additional cut min cx s. t. Qλ x = 0 1λ λ Ax = b = 1 i a x 0 x 0 New dual variable µ clearly do = not change the pricing. i b i

19 Dantzig-Wolfe X Explicit Master Lower bound L Lemma: Z DWM = Z EM Sketch of the Proof: Let λ and ( µ, ν ) be optimal primal and dual solutions of DWM. Then ( λ, x = Qλ ) and ( π = ( µ A c), µ, ν ) are optimal primal and dual solutions of EM.

20 Dantzig-Wolfe X Explicit Master LP size L Let Q Q be the current n p submatrix of priced columns and let m be the current number of rows in A. Rows Cols Non-zeros DWM m + 1 p nz( AQ ) + p EM m + n + 1 p + n nz( A) + nz( Q ) + p + n DWM usually leads to smaller LPs. However, upper bounds on x variables have to be represented as rows in DWM.

21 Dantzig-Wolfe X Explicit Master Column generation convergence L As usually n > m, there are more dual variables to handle in EM. This may lead to slower convergence. It can be counterbalanced by proper stabilization.

22 Is there any advantage on working on the Explicit Master? L BCP easier to code. (Not a really good argument... ) EM allows cheap fixing of x variables by reduced costs. EM allows better multiple column generation.

23 0 j Fixing variables on DWM After solving DWM, choose a variable x j from the original formulation and solve: V ( µ, ν ) = min( c µ A) x + ν s. t Dx d m x N, x 1 Z + V µ ν Z x j 0 If DWM j (, ) INC, then j can be fixed to 0. (This technique is known as fixing by Lagrangean bound). L Depending on the subproblem structure, each attempt to fix a variable can be quite expensive.

24 Fixing variables on EM After solving EM, just check if Z + c j Z. EM INC L The diference between EM and DWM is better understood by having a look at the dual of EM. Max ν + µ b πq + ν1 π + µ A 0 c

25 Fixing variables on EM The dual of EM is equivalent to Max ν + µ b πq + ν1 0 π + µ A + c = c c 0 L By solving DWM one always gets a solution of form ( π = ( µ A c), µ, ν ). This implies c = 0, no variable is ever fixed. In other words, solving DWM is equivalent to solving EM with the dual space restricted by c = 0 constraints.

26 Fixing variables on EM The fixing can much improved by perturbing the dual of EM Max ν + µ b + cε π Q + ν1 0 π + µ A + c = c c 0 This is equivalent to min cx s. t. Qλ x = 0 Ax = b 1λ λ = 1 0 x ε L

27 Multiple Column Generation Let (O) be the following IP with n variables: Min cx Ax (O) Dx Fx x = b d f N n { } n and suppose P1 = x N Dx d and are finite sets. { n } P = x N Fx f 2

28 Multiple Column Generation Min cx S.t. Q λ x = λ λ 1 1 Q λ 2 2 1λ λ 2 2 x Ax = = = = b x Min S.t. cx Ax = b { m } { m } x Conv x N Dx d x Conv x N Fx f EM allows multiple sets of constraints to be convexified independently. On DWM this would be cumbersome.

29 Dantzig-Wolfe X Explicit Master Implementing a BCP over the EM is interesting when: L Fixing of variables is important and the subproblem is too hard for effective Lagrangean fixing. Multiple column generation is performed.

30 The symmetry problem On many important problems where BP is applied, the corresponding original formulation is highly symmetric. Some examples are: L bin packing cutting stock problems graph coloring clustering problems scheduling with some identical machines

31 The symmetry problem Most of those problems share the same characteristic: They search for an optimal feasible assignment of n elements to m sets, and some of those sets are undistinguishable. The original formulation corresponding to their usual k column generation reformulation have variables like x i, meaning that element i is assigned to set k. Cutting (and even branching) over those variables is not effective. L

32 How to construct a robust BCP for those problems? We do not have a full answer to this, just an approach that seems reasonable. L Ryan and Foster (1981) proposed a branch rule widely used on BPs for those problems. In one branch, elements i and j must be in the same set; on the other branch they must be in different sets. Valério de Carvalho (2001) suggested solving binpacking by BP considering it as a particular case of the VRPTW.

33 How to construct a robust BCP for those problems? L Those ideas inspire a general approach: 1. Formulate the original problem over variables meaning that elements i and j are assigned to the same set. Apply IP reformulation. 2. Is the resulting pricing problem still tractable? Go ahead. 3. Find additional families of valid inequalities over the original formulation. x ij

34 Example: vertex coloring problem BP algorithm (Mehotra and Trick, 1996) Pricing problem: maximum weight independent set over G. Branch rule: Ryan and Foster. Corresponding original formulation: variables, vertex i is colored k. Not good for BCP. Set-partition with a row for each vertex and colunms are independent sets. Z EM =2.5 7 k x i L 6 4 3

35 Example: vertex coloring problem Possible BCP algorithm Pricing problem: maximum edge weight clique over. Branch rule: over any constraints on variables. Cuts: Odd holes, anti-odd holes, etc. Set-partition with a row for each vertex and colunms are vertex-sets corresponding to cliques in G, ZEM =2.5. Adding odd-hole cut x + x + x + x + x Z EM = x ij 3 G 2 L

36 Computational Results We are implementing robust BCPs for several classical problems. We already have results on the following problems: L CVRP CMST GAP

37 Capacitated Vehicle Routing Problem BP is known to be effective on tightly constrained VRP problems, like VRPTW (Desrosiers, Soumis and Desrochers, 1984). L BP does not work well on CVRP. Best exact algorithms are sophisticated BCs. They can solve instances with up to vertices.

38 Capacitated Vehicle Routing Problem Our BCP (Fukasawa, Poggi de Aragão, Reis, Uchoa) Dantzig-Wolfe Master. Columns correspond to q-routes without 2-cycles (Christofides, Mingozzi, Toth, 1981). Pricing is cheap (O(n 2 C)). Only capacity cuts. Consistently solves instances with up to 100 vertices, including almost all open instances from the literature. Better results are expected soon by adding several other cuts used by current BC codes.

39 Capacitated Minimum Spanning Tree Problem Two kinds of instances: unitary and non-unitary capacities. L Best exact algorithms: Unitary case: BC over large hop formulations. Non-unitary case: BC and Lagrangean relaxation. OR-Lib instances with only 50 vertices can not be solved.

40 Capacitated Minimum Spanning Tree Problem Our BCP (Fukasawa, Poggi de Aragão, Porto, Uchoa) Explicit Master. Columns correspond to connected components from the root. Pricing is expensive (O(n 2 C 2 )). Capacity and root cutset cuts. Results are good on the unitary case and very good on the non-unitary case: all open instances with 50 vertices and some with 100 vertices solved.

41 Generalized Assignment Problem Most instances from the OR-Lib are now considered to be easy and can be solved by commercial MIP packages. Only the D series remains challenging. L Best exact algorithms: Lagrangean relaxation, BP and BC.

42 Generalized Assignment Problem Our BCP (Pigatti, Poggi de Aragão, Uchoa) Dantzig-Wolfe Master. Pricing is cheap, knapsack problems. Fischetti-Lodi cuts. Good results. Could solve 3 out of 5 open instances from D series. The use of other cuts from the literature is being studied. Real world instances with symmetry are being considered.

43 Perspectives The implementation of robust BCP algorithms today is still quite restricted to the column generation community. The good results that are being obtained indicate that this situation may soon change. This certainly raises a number of interesting research topics.

44 Perspectives Obtaining high performance from a BC or from a BP may require experience to choose among several possible strategies. On a BCP the number of possibilities increases and little experience is currently available to make decisions. For example: Should I price to optimality before cutting? Or cutting should be done along with (or even before) pricing?

45 Perspectives For obvious reasons, polyhedral research concentrates on problems where BC works. BCP gives motivation to studying other polytopes. For example: What are the families of facets of the bin-packing polytope over the variables? x ij In a similar way, robust BCP motivates to devise new column generation schemes on problems where BP alone would not work.