Stabilization in Column Generation: numerical study

Transcription

1 1 / 26 Stabilization in Column Generation: numerical study Artur Pessoa 3 Ruslan Sadykov 1,2 Eduardo Uchoa 3 François Vanderbeck 2,1 1 INRIA Bordeaux, France 2 Univ. Bordeaux I, France 3 Universidade Federal Fluminense, Brazil ROADEF 2013 Troyes, France, February 13

2 2 / 26 Contents Introduction Stabilization Techniques Numerical tests

3 3 / 26 Problem decomposition Assume a bounded integer problem: [F] min c x : A x a x Z = { B x b x N n } Assume that subproblem [SP] min{c x : x Z } (1) is relatively easy to solve compared to problem [F]. Then, Z = {z q } q Q conv(z ) = {x R n + : q Q z q λ q, q Q λ q = 1, λ q 0 q Q}

4 4 / 26 Lagrangian Relaxation & Duality L(π) := min q Q {c zq + π (a Az q )} [LD] := max min {c π R m + q Q zq + π(a Az q )} η η η L(π) L(π) π π π

5 5 / 26 Lagrangian Dual as an LP [LD] max min {π a + (c π R m + q Q πa)zq }; max{η, η cz q + π(a Az q ) q Q, π R m +, η R 1 }; min{ q Q(cz q )λ q, η (Az q )λ q a, q Q λ q = 1, q Q λ q 0 q Q}; min{cx : Ax a, x conv(z ) }. L(π) π

6 6 / 26 Dantzig-Wolfe Reformulation & Restricted Master min q Q(cx)λ q (Az q )λ q a q Q λ q = 1 q Q λ q {0, 1} q Q. [M t ] min{ q Q t cz q λ q : q Q t Az q λ q a; q Q t λ q = 1; λ q 0, q Q t } [DM t ] max{η : π(az q a) + η cz q, q Q t ; π R m +; η R 1 }

7 7 / 26 Restricted Master, Dual Polyhedra, & Pricing Oracle [M t ] min {cx : Ax a, x conv({z q } q Q t )}. η L t () : π L t (π) = min q Q t {πa + (c πa)z q }; Solving [LSP(π t )] yields: 1. most neg. red. cost col. for [M t ] 2. most violated constr. for [DM t ] 3. a sub-gradient g t = (a A z t ) of L(.) 4. the correct value of L(.) at point π t L(π) (π t, η t ) π

8 8 / 26 Dual Polyhedra: Outer and Inner approximations η η (π t, η t ) (ˆπ, ˆL) L(π) L(π) π π

9 9 / 26 Convergence of Column Generation A sequence of candidate dual solutions {π t } t π A sequence of candidate primal solutions (a by-product) {x t } t x Dual oscillations Tailing-off effect Primal degeneracy

10 10 / 26 Contents Introduction Stabilization Techniques Numerical tests

11 Penalty functions (ˆπ, ˆL) { } π t = argmax L t (π) Ŝt(π) π R m + L(π) Ŝ(π ˆπ) min q Q t cz q λ q + ˆπ ρ + Ŝ t (ρ) [ M t ] Az q λ q + ρ a q Q t λ q = 1 q Q t q Q t λ q 0 max π a + η Ŝt(π ˆπ) πaz q + η c z q [ DM t ] q Q t (π, η) R m + R 1 11 / 26

12 12 / 26 Piecewise linear penalty functions 3-pieces: [du Merle, Villeneuve, Desrosiers, Hansen 99] 5-pieces: [Ben Amor, Desrosiers, Frangioni 09] ε out ˆπ in ε in π out 4 additional variables per master constraint.

13 13 / 26 Dual Price Smoothing (I) π t = αˆπ + (1 α)π t [Wentges 97] (π in, η in ) := (ˆπ, ˆL) (π out, η out ) := (π t, η t ) (π sep, η sep ) := α (π in, η in ) + (1 α) (π out, η out ) OUT SEP IN

14 14 / 26 Dual Price Smoothing (II) Case A: Case B: Case C: π in SEP is cut, so is OUT SEP is not cut, but OUT is cut neither SEP nor OUT is cut mis-price π 1 π 2 π 3 π out OUT SEP IN OUT OUT SEP IN SEP IN

15 15 / 26 Smoothing with a static α (I) Generalized assignment OR-Library C, D, E instances: 10 jobs per agent 100 jobs / 10 agents; 200 jobs / 20 agents; 400 jobs / 40 agents 600 Master iterations Subproblem calls 100 Time (sec.) smoothing parameter (α) smoothing parameter (α)

16 16 / 26 Smoothing with a static α (II) Generalized assignment OR-Library C, D, E instances: 40 jobs per agent 200 jobs / 5 agents ; 400 jobs / 10 agents 5, , ,000 2, ,000 0 Master iterations Subproblem calls smoothing parameter (α) Time (sec.) smoothing parameter (α)

17 17 / 26 Smoothing: auto-adaptative α-schedule π out π g( π) decrease α increase α L(π) = ˆL π in

18 18 / 26 Directional smoothing hybridization with ascent methods π out L(π) = ˆL π α π in π out β π out π g γ π sep π in g in π g Automatic directional smoothing: β = cos γ

19 19 / 26 Contents Introduction Stabilization Techniques Numerical tests

20 20 / 26 Test problems Parallel Machine Scheduling: 30 instances generated in the same way as in the OR-Library with number of machines in {1, 2, 4} and jobs in {50, 100, 200}. Generalized Assignment: 18 OR-Library instances of types D and E with number of agents in {5, 10, 20, 40} and jobs in {100, 200, 400}. Multi-Echelon Small-Bucket Lot-Sizing: 17 randomly generated instances varying by the number of echelons in {1, 2, 3, 5}, items in {10, 20, 40}, and periods in {50, 100, 200, 400}. Bin Packing: 12 randomly generated instances with number of items in {400, 800} and average number of items per bin in {2, 3, 4}. Capacitated Vehicle Routing: 21 widely used instances from the literature of types A, B, E, F, M, P with clients and 4-16 vehicles.

21 21 / 26 Auto-adaptative Wentges Smoothing It number of iterations in column generation Col max. number of columns in master Tim solution time Geometric means are shown Ratio α = 0 Ratio α = 0 vs α = best vs α = auto Problem It Col Tim It Col Tim Generalized Assignment Lot-Sizing Machine Scheduling Bin Packing Vehicle Routing

22 22 / 26 Directional Wentges Smoothing Ratio α = best, β = 0 Ratio α = best, β = 0 vs α = best, β = best vs α = auto, β = auto Problem It Col Tim It Col Tim General. Assignment Lot-Sizing Machine Scheduling Bin Packing Vehicle Routing

23 Penalty function setup Hard to cover all parameter setting we concentrate on a good class of symmetric 5-piece linear functions: ε ext = 0.9 ˆπ int = 0.2 ext ext ε int = 0.1 int is multiplied by κ each time the stability center changes ε int is divided by 3 each time a artificial variable in the optimal solution Exhaustive search for best couple of parameters ext, κ (instance dependent) π 23 / 26

24 24 / 26 Smoothing vs. Penalty function stabilization Are shown ratios of non-stabilized column generation versus x Smoothing is with α = auto, β = auto x It Col Tim Machine Scheduling Smoothing auto Penalty function tuned Penalty function + smoothing tuned Generalized assignment Smoothing auto Penalty function tuned Penalty function + smoothing tuned

25 25 / 26 Instance GAP - D - 10agents - 400jobs 6000 it col 5 h Non-stabilized column generation Wentges smoothing 500 it col 8 m 161 it col 15 s Penalty function Wentges + directional smoothing 309 it col 2 m 50 s 209 it col 12 s Pen. func. + Wentges smoothing 4 500x speed-up 122 it col 4 s Pen. func. + smoothing + dir. smooth.

26 26 / 26 Instance GAP - D - 10agents - 400jobs 6000 it col 5 h Non-stabilized column generation Wentges smoothing 500 it col 8 m 161 it col 15 s Penalty function Wentges + directional smoothing 309 it col 2 m 50 s 209 it col 12 s Pen. func. + Wentges smoothing TO DO: automize this! 122 it col 4 s Pen. func. + smoothing + dir. smooth.