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STABILIZED COLUMN GENERATION

STABILIZED COLUMN GENERATION. Jacques Desrosiers HEC & GERAD. Hatem Ben Amor François Soumis Daniel Villeneuve GERAD. Multiple Depot Vehicle Scheduling Problem. MDVSP. Motivation. Using Optimal Dual Values. Optimal Dual Values.

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STABILIZED COLUMN GENERATION

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  1. STABILIZED COLUMN GENERATION Jacques Desrosiers HEC &GERAD Hatem Ben Amor François Soumis Daniel Villeneuve GERAD

  2. Multiple Depot Vehicle Scheduling Problem

  3. MDVSP

  4. Motivation

  5. Using Optimal Dual Values

  6. Optimal Dual Values • Useful in the context of Lagrangian Relaxation to recover primal feasibility • Useful to perform crossover from an interior point solution to an extreme point solution

  7. Crossover: CPLEX vs STABILIZATION (cpu times in seconds)

  8. MASTER PROBLEM ColumnsDual Multipliers COLUMN GENERATORS (Shortest Path Problems on Acyclic Graphs) LP Column Generation Optimality Conditions: primal feasibility dual feasibility complementary slackness

  9. 1960 Gillmore & Gomory Cutting Stock Problem 1989 Agarwal, Mathur & Salkin VRP 1963 Marquardt Trust region 1975 Marsten Box Step 2000 Kallehauge & Madsen VRPTW 1992 Vial & Goffin ACCPM 1999 du Merle et al. Stabilized CG 2000 Valério de Carvalho Cutting Stock Problem Some References

  10. Impact of Dual Boundson the Primal Formulation Primal (P) Dual (D) Restricted Dual (Dd) A Relaxed Primal (Pd)

  11. Trust Regions / Box Step b

  12. Dual Cutsfor the Cutting Stock problem Valério de Carvalho (2000) “Using extra dual cuts to accelerate column generation” Small items (i=1,…,m) are ranked : Additionally: Using a priori at most 2m dual constraints (or primal columns) ...

  13. Dual Cuts / Primal ColumnsValério de Carvalho (2000) Application to the Cutting Stock Problem reduces the CPU time by 40%.

  14. Triplets (501 items)each roll of length L is cut into exactly three orders without any waste

  15. Degeneracy & Perturbation Primal (P) A relaxed primal Perturbed Primal (Pe)

  16. Stabilized Column Generation Restricted Dual Perturbed Primal Stabilized Primal ( )

  17. Stabilized Primal & Dual Stabilized Primal ( ) Stabilized Dual ( )

  18. Dual Space b>0

  19. Propositions

  20. Propositions

  21. Parameter Adjustment, by component b>0 If the dual value is too small, re-center and enlarge the interval. If dual value is within the interval, re-center and reduce the interval. If the dual value is too large, re-center and enlarge the interval.

  22. delta=100 + network pi + network solution delta=100 + network pi Increasing lower bound pi*b

  23. 5 linear pieces + network pi + network solution + direction of “b”

  24. 5 linear pieces + network pi + network solution + direction of “b”

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