Column Generation for Extended Formulations

Transcription

1 1 / 28 Column Generation for Extended Formulations Ruslan Sadykov 1 François Vanderbeck 2,1 1 INRIA Bordeaux Sud-Ouest, France 2 University Bordeaux I, France ISMP 2012 Berlin, August 23

2 2 / 28 Contents Motivation Methodology Interest of the approach Numerical results and conclusions

3 3 / 28 Extended formulations Reformulation involving extra variables tighter relations between variables Ways to obtain Variable Splitting (binary or unary expansion) Network Flow (Multi-Commodity) Dynamic Programming Solver [Martin et al] Union of Polyhedra [Balas] Polyhedral Branching Systems [Kaibel, Loos]...

4 4 / 28 Ways to exploit extended formulations 1. Use a direct MIP-solver approach: size is an issue. 2. Use projection tools: Benders cuts. dynamic outer approximation of the formulation 3. Use of an approximation [Van Vyve & Wolsey MP06] Drop some of the constraints Aggregate commodities Partial reformulation static outer approximation of the formulation 4. Use (delayed) column generation. dynamic inner approximation of the formulation

5 5 / 28 Column-and-row generation It is a generalization of the standard column generation (based on the Dantzig-Wolfe reformulation). Our contributions Reviewing of the methodology of the column-and-row generation and presenting it as a generic approach Analysis of the interest of the column-and-row generation approach: its good performance is explained by a stabilization effect New computational results

6 6 / 28 Contents Motivation Methodology Interest of the approach Numerical results and conclusions

7 Extended formulation for a subsystem Original formulation { [F] min c x A x a B x b x Z n + } Subsystem { P B x b x R n + X = P Z n } Main assumption There exists a polyhedron Q = { Hz h, z R e } + and transformation T s.t. Q defines an extended formulation for X: { } conv(x) = proj x Q = x = Tz : Hz h, z R e + 7 / 28

8 8 / 28 Extended reformulation Original formulation { [F] min c x A x a B x b x Z n + } Extended reformulation { [R] min c T z A T z a H z h z Z e + } Special case: Dantzig-Wolfe reformulation { [M] min c x g λ g g G A x g λ g a g G λ g = 1 g G λ {0, 1} G } x 1 x 2 x 4 x 3

9 9 / 28 Column-and-row generation: a hybrid approach Alternative to direct resolution by a MIP solver Dynamic generation of the variables of [R]: generated in bunch by optimizing over X. Adding rows that become active. Alternative to the standard column generation Perform the column generation for [M] Project the master program in [R] (we split generated columns into individual variables)

10 10 / 28 Example: machine scheduling with a sum criterion S 3 3 S 2 2 S { [R] min c jt z jt jt T p j t=0 z jt = 1 j J z j0 = 1 j J z jt z j,t pj = 0 t 1 j J j J } z jt {0, 1} j, t t { min c(s j ) j S j + p j S } i (i, j) or S i + p i S j { [M] min c g λ g g G T p j g G t=0 z g jt λ g = 1 j J λ g = 1 g G } λ g {0, 1} g G

11 Machine scheduling: column-and-row generation 1. Solve the restricted extended formulation [R LP ] (start from a feasible one) and update dual prices. 2. Solve the pricing subproblem (obtain a pseudo schedule) t 3. Disaggregate the subproblem solution in arc variables z. z 30 z23 z If some of these variables z are not in [R LP ], add them to it along with the associated flow conservation constraints, then go to step Otherwise stop (the current solution of [R LP ] is optimal for [R]). 11 / 28

12 12 / 28 Restricted reformulations Z = {z s } s S a set of integer solutions of Q, S S z restriction of z to the components of s S supp(zs ) G = G(S) = {g G : x g = T z s, s S} { [R LP ] min c T z Proposition A T z a H z h z R e + } { [M LP ] min c x g λ g g G A x g λ g a g G v [M LP] = v [R LP] v [R LP] v [M LP]. λ g = 1 g G } λ R G +

13 Column-and-row generation procedure Step 0: Initialize the dual bound, β :=, and a subset S so that [R LP ] is feasible. Step 1: Solve [R LP ] and collect its dual solution π associated to constraints A T z a. Step 2: Obtain a solution z of the pricing problem: min{(c πa)t z : z Z } = min{(c πa)x : x X}. Step 3: Compute the Lagrangian dual bound: L(π) π a + (c πa) T z, and update β max{β, L(π)}. If v [R LP] β, STOP. Step 4: Update the current bundle S by adding solution z and update [R LP ]. Go to Step 1. Proposition Either v [R LP] β (stopping condition), or some of the components of z have negative reduced cost in [R LP ]. 13 / 28

14 14 / 28 Example: multi-item multi-echelon lot sizing yet k xet k setup for item k at echelon e in period t production for item k at echelon e in period t [F] { min (cet k xet k + fet k yet) k : ket yet k 1 k t xeτ k τ=1 t τ=1 e, t x k e+1,τ t xeτ k D1t k k, t τ=1 xet k DtT k yet k k, e, t xet k 0 k, e, t } yet k {0, 1} k, e, t k, e < E, t

15 Multi-echelon lot sizing: extended formulation Dominance property There exists an optimal solution in which x et s et = 0 k, e, t production plan for every item k is a directed tree: e = 1 e = 2 e = 3 t Dynamic programming State (e, t, a, b) corresponds to accumulating at echelon e in period t a production covering exactly the demand of periods a,..., b. Extended formulation follows from [Martin et al]. 15 / 28

16 16 / 28 A generalization Relaxed assumption There exists a polyhedron Q = { Hz h, z R e } + and transformation T s.t. Q defines a tighter formulation for X: { } conv(x) proj x Q = x = Tz : Hz h, z R e + P Consequences Column-and-row procedure is still valid However, in general, the dual bound is not as tight as v [M LP].

17 17 / 28 Contents Motivation Methodology Interest of the approach Numerical results and conclusions

18 18 / 28 Column-and-row generation vs. column generation Proposition reminder v [M LP] = v [R LP] v [R LP] v [M LP]. Remark Column-and-row generation can converge faster than the standard column generation. But when (and why) this happens? Recombination property Given S, subproblem solutions z 1,..., z k Z (S) can be recombined in a new solution ẑ [R LP ] such that ẑ conv(z (S)).

19 19 / 28 Machine scheduling: recombination property Z (S) = {z 1, z 2 }, ẑ [R LP ] z z ẑ 5 2 t

20 20 / 28 Machine scheduling: example of convergence Column generation for [M] Column-and-row generation for [R] Initial solution Iteration Subproblem solution Subproblem solution Final solution

21 21 / 28 Multi-echelon lot-sizing: recombination property Z (S) = {z 1, z 2 }, ẑ [R LP ] z 1 z 2 e = 1 e = 2 e = 3 e = 1 e = 2 e = 3 ẑ e = 1 e = 2 e = 3

22 22 / 28 Contents Motivation Methodology Interest of the approach Numerical results and conclusions

23 Machine Scheduling: numerical results Generated similarly to the instances from the OR-library Averages for 25 instances are given Processing times are in [1,..., 100]. Cplex 12.1 Colomn gen. Column-and-row for [R LP ] for [M LP ] generation for [R LP ] m n cpu #it cpu #it vars cpu % % % % % % % % >2h % 39.4 #it number of column generation iterations vars percentage of variables z generated cpu solution time, in seconds 23 / 28

24 Machine Scheduling: results with smoothing Both column and column-and-row generation are stabilized with smoothing: pricing problem is solved for the vector of dual values which is a linear combination of current dual solution and the stability center (smoothing parameter α is the best possible). Colomn gen. Column-and-row gen. for [M LP ], α = 0.9 for [R LP ], α = 0.5 m n #it cpu #it vars cpu % % % % % % % % % / 28

25 25 / 28 Multi-echelon lot sizing: results with smoothing Averages for 10 instances are given Colomn gen. Column-and-row for [M LP ], α = 0.85 gen. for [R LP ], α = 0.4 E K T #it cpu #it vars cpu % % % % % % % % % % % % 386.1

26 26 / 28 Conclusions 1. Column generation for an extended formulation is to be considered when: The extended formulation is obtained using a decomposition. SP solutions can be recombined into alternative ones. 2. The approach can be interpreted as a stabilization method for column generation: disaggregation helps, related to the use of exchange vectors, combined effect with other stabilization techniques (e.g. smoothing). 3. Computational results (ours and in the literature) show that this can be a competitive approach.

27 27 / 28 Bin Packing: results with smoothing Bin capacity is 4000 Item sizes are generated uniformly in intervals [1000, 3000] ( a2 ), [1000, 1500] ( a3 ), and [800, 1300] ( a4 ) Averages for 5 instances are given Cplex 12.1 Col. gen. Col-and-row gen. for [F] for [M], α = 0.85 for [R], α = 0.85 class n gap %gap cpu #it cpu #it cpu vars a a a

28 28 / 28 Generalized Assignment: results with smoothing Instances from the OR-Library (class D) Cplex 12.1 Col. gen. Col-and-row gen for [F LP ] for [M LP ], α = 0.85 for [R LP ], α = 0.5 m n %gap cpu #it %gap cpu #it %gap cpu vars