Decomposition and Reformulation in Integer Programming

Transcription

1 and Reformulation in Integer Programming Laurence A. WOLSEY 7/1/2008 / Aussois and Reformulation in Integer Programming

2 Outline 1 Resource 2 and Reformulation in Integer Programming

3 Outline Resource 1 Resource 2 and Reformulation in Integer Programming

4 Implicit Pricing for LP Resource Ford and Fulkerson (1958) "A Suggested Computation for Maximal Multicommodity Flow" Use an arc-path formulation. To price out the large number of columns, they propose to take the LP dual variables π ij on the arcs, and then solve the subproblem of finding a shortest source-sink path with lengths π ij for each commodity k. Dantzig-Wolfe (1960) and Reformulation in Integer Programming

5 Resource Cutting Stock Problem. Gilmore and Gomory (1961) Standard lengths L. Requirement w i pieces of length l i for i = 1,..., m Each pattern is a column: J z = min x j : j=1 J P j x j w, x Z J + j=1 Solve the LP relaxation. Column generation subproblem is now a knapsack problem. max{ j π jx j : j l jx j L, x Z m +} Discussion of rounding the LP solution to get a good IP solution, etc. (1963) Multiple stock lengths, knife limitations, etc. and Reformulation in Integer Programming

6 Multi-Item Lot-Sizing Resource Dzielinski and Gomory (1965) Simplified Model: min i (ci x i + hs i + q i y i ) st 1 i + x t i = dt i + si t, x t i Myt i i, t i (aik yt i + bik xt i) W t k k, t x, s 0, y {0, 1} IT Structure of Wagner-Whitin (1958) optimal solutions known: original reformulation min Cj iθi j i j (aik yj,t i + bik xjt i )θi j Wt k j θi j = 1 i θ i j 0 i, j k, t and Reformulation in Integer Programming

7 Dzielinski and Gomory (cont) Resource Column generation subproblem is a Wagner-Whitin lot-sizing problem. LP does not give integer solutions, but at most KT θ i j are non-integer. Size envisaged: K = 10, T = 10, I = 500 KT + I rows Work with one convexity constraint. Reduce to KT + 1 rows. Observations relative to Long Tail. and Reformulation in Integer Programming

8 Dzielinski and Gomory Resource Typical instance I = 428, T = 7, K = 2,S = 3 shifts First formulation 486 rows and columns Reformulation: 59 rows, 144 columns Time 10 minutes, 52 iterations Trigeiro et al. (1987) T = 20, I {10, 20, 30}, K = 1 MIPs still cannot be solved in 10 minutes, or even hours, even though the tight LP can be solved in less than a second. and Reformulation in Integer Programming

9 Towards Branch-and-Cut-Price Resource Branch on original variables if possible (i.e if 0-1) Ryan and Foster(1981) How to branch on the θ variables. in set partitioning problems J. Desrosiers, F. Soumis and M. Desrochers, (1984) and onwards. Carry out column generation at each node of the branch-and-bound tree. Many applications in routing. Stabilization/Bundle methods to give faster convergence. Comparison of bundle and classical col. generation (Bryant et al., MPA 2007) Selected Topics in Col. Generation (Lübbecke and Desrosiers Oper. Res. 2005) and Reformulation in Integer Programming

10 Branch-and-Cut-and-Price Resource min{cx : x Y Z } Reformulation min cx x = i I λ ix i, i I λ i = 1, λ 0 conv(y ) π j x π j 0 j J conv(z ) x Z n + Example: Robust Branch-and-Cut-and-Price for CVRP (Fukosawa et al. MPA 2006) conv(y ) columns are q-routes. conv(z ) rows are capacitated subtour elimination constraints and Reformulation in Integer Programming

11 Resource Generalized Lagrange Multiplier Method Everett (1963) Report based on several applications: Problem max{c(x) : g(x) b, x X} or max{ k ck (x k ) : k gk (x k ) b, x k X k, k = 1,..., K } Several results: If λ R m + and x is optimal in max{c(x) i λ ig i (x) : x X}, then x is optimal in max{c(x) : g(x) g(x ), x X}. Problem 2 breaks up into small independent subproblems. Obtains the convex envelope of the value function. and Reformulation in Integer Programming

12 Resource and Reformulation in Integer Programming

13 Resource The traveling salesman problem. Held and Karp 1970 A 1-tree is a spanning tree on nodes {2,..., n} plus two edges incident to node 1. Every tour is a 1-tree. A minimum cost 1-tree can be found by greedy or other. Dual subproblem: z(u) = min{ ij (c ij u i u j )x ij + 2 i u i : x X 1 Trees } Lagrangian dual: z LD = max u z(u) The lagrangian dual based on the 1-tree relaxation is equivalent to solving the subtour LP. Other equivalent LP formulations. and Reformulation in Integer Programming

14 Resource STSP and Subgradient Algorithm. Held,Karp (1971) and Reformulation in Integer Programming

15 Resource Lagrangean Relaxation for IP Geoffrion (1974) (P) z = min{cx : Ax b, x X} Dual subproblem (P(u)) z(u) = min{cx + u(b Ax) : x X} Lagrangean dual (LD) w LD = max u 0 z(u) Main Theorem w LD = min{cx : Ax b, x conv(x)} Integrality Property! and Reformulation in Integer Programming

16 And then? Resource Dual ascent heuristics Uncapacitated Facility Location: Bilde and Krarup(1977) Erlenkotter (1978) Directed Steiner Tree: Wong (1984) Multicommodity (Uncapacitated) Fixed Charge Network Flows: Balakrishnan et al. (1989) Improvements to the Subgradient Algorithm Bundle Methods Lagrangian : Jornsten and Nasberg(1986), Guignard and Kim (1987) and Reformulation in Integer Programming

17 Benders (1962) Resource Partitioning Procedures for Solving MIPs max{cx + hy : Ax + Gy b, x Z n +, y R p + } Reformulation η k = max η η u t (b Ax) + cx t T k 0 v s (b Ax) s S k η R 1, x Z n + Subproblem: ζ k = max{hy : Gy b Ax k, y R p + } If infeasible, add an infeasibility cut: v s (b Ax) 0. If feasible with ζ k + cx k > η k, add an optimality cut η u t (b Ax) + cx. If feasible with ζ k + cx k = η k, Optimal. and Reformulation in Integer Programming

18 Generalizations and Extensions Resource Geoffrion Elements of Large Scale Math Prog.(1970) Geoffrion and Graves (1974) Multicommodity Distribution System Design. Commodities, Plants, Distribution Centers (DCs) and Customer zones (CZs). Fix open DCs and assignment each CZ to one DC. Subproblem: Transportation problem for each item Only 3-5 major iterations Applications of Benders - stochastic programming, not too many MIPs. Folklore: Too many costly iterations. "Strong" Benders cuts: Magnanti and Wong (1981) and Reformulation in Integer Programming

19 Recent Resource Benders by Branch-and-Cut: The subproblem just provides a separation algorithm for each node. If LP solution is integer, one must call the separation routine to be sure of feasibility and/or optimality. If LP solution is fractional, one may call the separation routine or branch. Branch-and-Check: Integer Subproblems where one just checks feasibility. Add infeasibility cuts, i.e. if x {0, 1} j:x j =0 x j + j:x j =1 (1 x j ) 1. Laporte and Louveaux (1993), Jain and Grossman(2001) Integer subproblems remain a big challenge. and Reformulation in Integer Programming

20 Outline 1 Resource 2 and Reformulation in Integer Programming

21 Questions When did the importance of tightening formulations become apparent? When did the importance of reformulations become apparent? When did one start looking for convex hulls? When did one start looking for extended (compact and tight) formulations? Are there systematic ways to develop tight extended formulations? What are the limits? and Reformulation in Integer Programming

22 : UFL Uncapacitated Facility Location - Disaggregation m x ij my j, x ij [0, 1] i, y j {0, 1} j=1 replaced by x ij y j, x ij [0, 1] i, y j {0, 1} P 2 P 1, so tighter bound. Now more precisely P 2 = conv(p 1 ([0, 1] mn {0, 1} n )). and Reformulation in Integer Programming

23 : 0-1 Knapsacks Equivalent 0-1 Knapsacks {x {0, 1} n : n a j x j b} j=1 replaced by an equivalent constraint with minimal integer coefficients. Williams (1974), Kianfar (1973), Bradley et al.(1974) 6x x x x x 5 45, x {0, 1} 5 x 1 + 2x 2 + 2x 3 + 3x 4 + 4x 5 5, x {0, 1} 5 Cover inequalities, etc Balas, Hammer et al.(1975) Crowder, Padberg, Johnson (1983) and Reformulation in Integer Programming

24 What is an extended formulation? P = {x R n + : Ax b} Q = {(x, w) R n + R p + : Cx + Dw e}. Q is an extended formulation for P if proj x (Q) = P, i.e. x P if and on;y if there exists w with (x, w) Q. If P = conv(x), then proj x (Q) = conv(x) and Reformulation in Integer Programming

25 : Examples Polyhedra are finitely generated (Minkowski) Union of Polyhedra (Balas 1975) Uncapacitated Lot-Sizing (Bilde and Krarup 1977) Multicommodity Reformulation of Fixed Charge Network Flows (Rardin and Choe 1979) STSP Multicommodity Reformulation (Wong 1980) Perfectly Matchable Subgraph of a Bipartite Graph (Balas and Pulleyblank 1983) Lift and Project, etc Balas et al.(1993), Sherali and Adams(1990), Lovasz and Schrijver(1991) and Reformulation in Integer Programming

26 Uncapacitated Lot-Sizing as a UFL Let w ut be fraction of demand in t produced in u. Production periods are locations. Demand periods are clients. min u p ux u + t f ty t t u=1 w ut = 1 for t = 1,..., n w ut y u for 1 u t n w 0, y {0, 1} n x u = n t=u d tw ut for u = 1,..., n LP relaxation solves Uncap. Lot-Sizing (Bilde-Krarup 1977) and Reformulation in Integer Programming

27 Single Source Fixed Charge Network Flow. Rardin j x 1j j x j1 = n i=1 b i j x ij j x ji = b i 0 i = 2,..., n x ij My ij (i, j) A x R m +, y {0, 1} m T = {i V : b i > 0}. Each k T gives a new commodity. j w k 1j j w k j1 = b k k T j w k ij j w k ji = b k δ ik i = 2,..., n, k K w k ij b k y ij (i, j) A, k A x ij = k w k ij (i, j) A x R m +, y {0, 1} m and Reformulation in Integer Programming

28 TSP and Subtour Polytope Miller, Tucker, Zemlin (1960). u i is position of node i in tour. Gavish and Graves (1978). Open arcs allow a flow from node 1 to every other node implying connectedness Wong (1980). Multi-commodity flow for each pair (p, q) in (1, i) and (i, 1) for i {2,..., n}. j y pq ij j y pq ji = δ ip + δ iq (i, j) A, (p, q) 0 y pq ij x ij (i, j) A, (p, q) j x ij = 1 i V i x ij = 1 j V y 0, x {0, 1} m Projection gives subtour elimination constraints! and Reformulation in Integer Programming

29 Using Extended Formulation OPTimization by LP Solve the LP max{cx + 0w : Cx + Dw e, x R n +, w R p + } Extended Formulation SEParation by LP Test by LP whether {w R p + : dw e Cx } =. Phase 1 of simplex (Farkas, Benders) gives cut. Extended Formulation Projection (Convex Hull) P = {x R p + : v t (e Cx) 0fort = 1,..., T } and Reformulation in Integer Programming

30 Finding OPTimization by DP Extended Formulation (Eppen and Martin ) SEParation by LP Extended Formulation (Martin 1991) No symmetric extended formulation for Matching and another intriguing result relative to the existence of extended formulations (Yannakakis 1991) and Reformulation in Integer Programming

31 OPTimization by DP Extended Formulation: DP for Uncapacitated Lot-Sizing G(0) = 0, G(j) = min i j (G(i 1) + f i + p i d ij ) This leads to the LP max{g(n) : G(j G(i 1) c i + p i d ij }. The dual LP is: min i j (f i + p i d ij )w ij i:i j w ij k:k>j w j+1,k = 0 j = 1,..., n 1 i:i n w in = 1 w ij 0 Finally set y i = j:j i w ij and x i = j:j i d ijw ij. and Reformulation in Integer Programming

32 Separation Extended Formulation for LS U Convex hull L = {1,..., l}, S L. x j + d jl y j d 1l. j S j L\S (x, y ) conv(x LS U ) iff l j=1 min[x j, d jl yj ] d 1l iff there exist α il, β il, z l such that max l [ i l (x i α il + d jl y j β il ) d 1l z l ] = 0 if and only if there exists π il such that α il + β il z l = 0 π jl xj, π jl d jl yj j, l l π jl d 1l l j=1 α, β, z 0 and Reformulation in Integer Programming

33 THE END May the festivities begin! and Reformulation in Integer Programming