Abstract
In this paper, we present a new perspective on cut generation in the context of Benders decomposition. The approach, which is based on the relation between the alternative polyhedron and the reverse polar set, helps us to improve established cut selection procedures for Benders cuts, like the one suggested by Fischetti et al. (Math Program Ser B 124(1–2):175–182, 2010). Our modified version of that criterion produces cuts which are always supporting and, unless in rare special cases, facet-defining. We discuss our approach in relation to the state of the art in cut generation for Benders decomposition. In particular, we refer to Pareto-optimality and facet-defining cuts and observe that each of these criteria can be matched to a particular subset of parametrizations for our cut generation framework. As a consequence, our framework covers the method to generate facet-defining cuts proposed by Conforti and Wolsey (Math Program Ser A 178:1–20, 2018) as a special case. We conclude the paper with a computational evaluation of the proposed cut selection method. For this, we use different instances of a capacity expansion problem for the european power system.
Highlights
Consider a generic optimization problem with two subsets of variables x and y where x is restricted to lie in some set S ⊆ Rn and x and y are jointly constrained by a setR
The interaction matrix H ∈ Rm×n captures the influence of the x-variables on the y-subproblem: For fixed x∗ ∈ Rn, (1.1) reduces to an ordinary linear program with constraints Ay ≤ b − H x∗, where A ∈ Rm×k, y ∈ Rk, and b − H x∗ ∈ Rm
We show that the alternative polyhedron can be viewed as an extended formulation of the reverse polar set, providing us with a parametrizable method to generate cuts with different well-known desirable properties, most notably facet-defining cuts
Summary
Consider a generic optimization problem with two subsets of variables x and y where x is restricted to lie in some set S ⊆ Rn and x and y are jointly constrained by a set. A cut generated from a point in the alternative polyhedron may be very weak, not even supporting the set epi(z) This is true even if we use a vertex of the alternative polyhedron and even if that vertex minimizes a given linear objective such as the vector 1 as suggested in Fischetti et al (2010). Our method is more robust with respect to the formulation of the problem than the original approach from Fischetti et al (2010) In particular it always generates supporting cuts, avoiding the problem pointed out in the context of Example 1.1. We obtain an (arguably simpler) alternative proof for the method to generate facet-defining cuts proposed by Conforti and Wolsey (2018), if applied to Benders decomposition. If γ minimizes γ b among all possible certificates in Theorem 1.2, the halfspace H(≤(π,π0),γ b) supports the set epi(z)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
