Ray projection for optimizing polytopes with prohibitively many constraints in set-covering column generation

Abstract

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 $${\mathbf r}\in \mathbb {Z}^n$$r?Zn, find the maximum $${t^*}\ge 0$$t??0 such that $${t^*}{\mathbf r}$$t?r is feasible. We interpret $${\mathbf 0_n}\rightarrow {\mathbf r}$$0n?r as a direction of growth from the origin $${\mathbf 0_n}$$0n; $${t^*}{\mathbf r}$$t?r is the intersection point between $${\mathbf 0_n}\rightarrow {\mathbf r}$$0n?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 $${\mathbf r}\in \mathbb {Z}^n$$r?Zn and (ii) $${\mathbf 0_n}$$0n is feasible. By iterating over such rays $${\mathbf r}$$r, our method produces a sequence of feasible solutions $${t^*}{\mathbf r}$$t?r 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).