On the mixing set with a knapsack constraint

TL;DR

This paper studies the mixing set with a knapsack constraint in mixed-integer programming, generalizing previous results from equal probability cases to the more complex general probabilities case, and provides polynomial-time separation algorithms.

Contribution

It characterizes new valid inequalities for the general probabilities case and extends previous results from equal probabilities, with efficient separation methods.

Findings

Characterization of inequalities not derived from the knapsack polytope.

Generalization of inequalities from the equal probabilities case.

Polynomial-time separation algorithms for a broad class of inequalities.

Abstract

The mixing set with a knapsack constraint arises as a substructure in mixed-integer programming reformulations of chance-constrained programs with stochastic right-hand-sides over a finite discrete distribution. Recently, Luedtke et al. (2010) and K\"u\c{c}\"ukyavuz (2012) studied valid inequalities for such sets. However, most of their results were focused on the equal probabilities case (equivalently when the knapsack reduces to a cardinality constraint), with only minor results in the general case. In this paper, we focus on the general probabilities case (general knapsack constraint). We characterize the valid inequalities that do not come from the knapsack polytope and use this characterization to generalize the inequalities previously derived for the equal probabilities case. We also show that one can separate over a large class of inequalities in polynomial time.