Sequence Independent, Simultaneous and Multidimensional Lifting of Generalized Flow Covers for the Semi-Continuous Knapsack Problem with Generalized Upper Bounds Constraints

Abstract

We consider the semi-continuous knapsack problem with generalized upper bound constraints on binary variables. We prove that generalized flow cover inequalities are valid in this setting. We also prove that, under mild assumptions, they are facet-defining inequalities for the full problem. We then focus on simultaneous lifting of pairs of variables. The associated lifting problem naturally induce multidimensional lifting functions, and we prove that a simple relaxation, in a restricted domain, is a superadditive function. We also prove that in many cases this approximation is actually the optimal lifting function. We then analyze the separation problem, which we separate in two phases: first, find a seed inequality, where we evaluate both exact and heuristic methods; secondly, since the lifting is simultaneous, our class of lifted inequalities might contain an exponential number of them. We choose a strategy of maximizing resulting violation. Finally, we test this class of inequalities on instances arising from electricity planning problems. Our test show that the proposed class of inequalities are strong in the sense that adding a few of these inequalities, they close, on average, 57.70% percent of the root integrality gap, and close 97.70% of the relative gap, while adding very few cuts.