Ray projection for optimizing polytopes with prohibitively many constraints in set-covering column generation
نویسنده
چکیده
A recurrent task in mathematical programming requires optimizing polytopes with prohibitivelymany constraints, e.g., the primal polytope in cutting-plane methods or the dual polytope in Column Generation (CG). This paper is devoted to the ray projection technique for optimizing such polytopes: start from a feasible solution and advance on a given ray direction until intersecting a polytope facet. The resulting intersection point is determined by solving the intersection sub-problem: given ray r ∈ Z, find the maximum t∗ ≥ 0 such that t∗r is feasible. We focus on dual polytopes associated to CG: if the CG (separation) sub-problem can be solved by Dynamic Programming (DP), so can be the intersection sub-problem. The convergence towards the CG optimum is realized through a sequence of intersection points t∗r (feasible dual solutions) determined from such rays r. Our method only uses integer rays r, so as to render the intersection sub-problem tractable by r-indexed DP. We show that in 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 only consists of fractional (or large-range) values. As such, the proposed method 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 that include Variable Size Bin Packing). We also prove the theoretical convergence of the proposed method.
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ورودعنوان ژورنال:
- Math. Program.
دوره 155 شماره
صفحات -
تاریخ انتشار 2016