On the mixing set with a knapsack constraint

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 Küçükyavuz (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.

An Introduction to Two-Stage Stochastic Mixed-Integer Programming

“…The separation problem of inequality (23) can be solved with sorting [26]. One may also use stronger inequalities than (23), as proposed in [1,29,36,70], however this is at the expense of harder separation problems.…”

A polyhedral study of the static probabilistic lot-sizing problem

“…Luedtke (2013), Küçükyavuz (2012) and Abdi and Fukasawa (2016) provide extensions of the basic mixing inequalities (5) for equal and general probability cases. However, the mixing inequalities based on cumulative production quantities do not provide any strengthening for fractional x.…”

Two-stage Stochastic Lot-sizing Problem with Chance-constrained Condition in the Second Stage

“…The first variant of SPLS is provided by Beraldi and Ruszczyski (2002), however, the objective function does not consider the inventory cost. Küçükyavuz (2012), Abdi and Fukasawa (2016) and Zhao et al (2017) all solve the SPLS model with the inventory costs in branch-and-cut algorithms, which are performed as the testification of the validity of their proposed valid inequalities for general chance-constrained programming problems. Another dynamic variant of SPLS that updates the production schedule after the scenario realization of the former time periods is studied by Zhang et al (2014).…”

“…Based on mixing set, Luedtke et al (2010) and Küçükyavuz (2012) give some stronger valid inequalities, and study under which conditions the proposed valid inequalities are sufficient to be facet-defining. Abdi and Fukasawa (2016) explore the characterization of valid inequalities for single mixing set, and explicitly develop a set of facet-defining inequalities under some particular conditions. Zhao et al (2017) generalize the valid and facet-defining inequalities presented in Küçükyavuz (2012) and Abdi and Fukasawa (2016), expect that another family of valid inequalities called knapsack cover inequalities is provided by lifting techniques.…”

“…Abdi and Fukasawa (2016) explore the characterization of valid inequalities for single mixing set, and explicitly develop a set of facet-defining inequalities under some particular conditions. Zhao et al (2017) generalize the valid and facet-defining inequalities presented in Küçükyavuz (2012) and Abdi and Fukasawa (2016), expect that another family of valid inequalities called knapsack cover inequalities is provided by lifting techniques. Zhou and Guan (2013) propose a two-stage stochastic lot-sizng problem (with backlogging), in which the planning horizon is separated into two stages, in the first stage the cost parameters are deterministic, while in the second stage the cost parameters are random variables, and the demands in whole horizon are deterministic.…”