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

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 ∈ Zn , 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 s...

• Type: Journal
• Year: 2016
• Where: MP
• Authors: Daniel Porumbel
• Keywords: Information Technology, MP 2016