[Por16a] Ray Projection for Optimizing Polytopes with Prohibitively Many Constraints in Set-Covering Column Generation

Revue Internationale avec comité de lecture : Journal Mathematical Programming, vol. 155(1), pp. 147-197, 2016, (

Auteurs: D. Porumbel

Mots clés: Linear Programming, Column Generation, Dynamic Programming, Ray Projection

Résumé: A recurrent task in mathematical programming consists of optimizing polytopes with prohibitively-many constraints, e.g., the primal polytope in cutting-planes methods or the dual polytope in Column Generation (CG). We present a method to optimize such polytopes using the following intersection sub-problem: given ray r∈Z_n , find the maximum t≥0 such that tr is feasible. We interpret 0→r as a direction of growth from the origin 0 ;tr is the intersection point between 0→r and the polytope boundary. To ease the exposition, the first part of the paper is devoted to the general context of a (primal or dual) polytope with rational data such that: (i) the intersection sub-problem can be effectively solved in practice for input rays r ∈ Zn and (ii) 0 is feasible. By iterating over such rays r, our method produces a sequence of feasible solutions tr that we prove to be finitely convergent. From Section 3 on, we focus on dual polytopes in CG formulations. Typically, if the CG (separation) sub-problem can be solved by Dynamic Programming (DP), so can be the intersection sub-problem. The proposed Integer Ray Method (IRM) only uses integer rays, and so, the intersection sub-problem can be solved using a DP scheme based on states indexed by integer ray values. We show that under such conditions, the intersection sub-problem can be even easier than the CG sub-problem, especially when no other integer data is available to index states in DP, i.e., if the CG sub-problem input consists of fractional (or large-range) values. As such, the IRM can tackle scaled instances (with large-range weights) of capacitated problems that seem prohibitively hard for classical CG. This is confirmed by numerical experiments on various capacitated Set-Covering problems: Capacitated Arc-Routing, Cutting-Stock and other three versions of Elastic Cutting-Stock (i.e., a problem class that includes Variable Size Bin Packing).

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@article {
title="{Ray Projection for Optimizing Polytopes with Prohibitively Many Constraints in Set-Covering Column Generation}",
author="D. Porumbel",
journal="Mathematical Programming",