From the separation to the intersection sub-problem in Benders decomposition models with prohibitively-many constraints

Abstract

We consider a minimization linear program over a polytope P described by prohibitively-many constraints. Given a ray direction 0 → r, the intersection sub-problem asks to find: (i) the intersection point t * r between the ray and the boundary of P and (ii) a constraint of P satisfied with equality by t * r. In [12, §2], we proposed a method based on the intersection sub-problem to optimize general linear programs. In this study, we use a Cutting-Planes method in which we simply replace the separation sub-problem with the intersection sub-problem. Although the intersection sub-problem is more complex, it is not necessarily computationally more expensive than the separation sub-problem and it has other advantages. The main advantage is that it can allow the Cutting Planes algorithm to generate a feasible solution (using t * r ∈ P) at each iteration, which is not possible with a standard separation sub-problem. Solving the intersection sub-problem is equivalent to normalizing all cuts and separating; this interpretation leads to showing that the intersection sub-problem can find stronger cuts. We tested such ideas in a Benders decomposition model with prohibitively-many feasibility cuts. We show that under certain (mild) assumptions, the intersection sub-problem can be solved within the same asymptotic running time as the separation one. We present numerical results on a network design problem that asks to install a least-cost set of links needed to accommodate a one-to-many flow. We associate each (u i , b i) ∈ C to a value t i = bi u i r > 0, so that t i r satisfies with equality the constraint u i y ≥ b i. Taking t * = max (ui,bi)∈C t i , we obtain t * ≥ t i for all (u i , b i) ∈ C, and so, u i (t * r) ≥ u i (t i r) = b i , where we used u i r ≥ 0. This shows that t * r satisfies all constraints in C, i.e., t * r ∈ P. We still need to show that t * is minimum with this property. This follows from the fact that t * r satisfies u i (t * r) = b i for some (u i , b i) ∈ C that maximizes (1.3). As such, any t < t * would lead to u i (tr) < b i , violating the constraint u i y ≥ b i. A similar result in the context of a maximization problem can be found in Proposition 3 of [12, §3.2]. Comparing (1.2) and (1.3), the intersection sub-problem can be seen as a generalized version of the separation sub-problem. We will also discuss in Section 2.5.1 that solving the intersection sub-problem is equivalent to normalizing all constraints (i.e., make them all have a right-hand side term of 1) followed by choosing one constraint by classical separation. Since the invention of the Benders decomposition in 1962 [2], the approach has become increasingly popular in optimization and hundreds of papers have used it for a wide variety of applications. 3 In particular, the Benders reformulation has been very successful for network design problems [5, 6, 7, 8]. The prohibitivelymany constraints C correspond to the extreme solutions (optimality cuts) and the extreme rays (feasibility cuts) of a polytope P referred to as the Benders sub-pro...