Approximate Submodularity and Its Implications in Discrete Optimization

Submodularity, a discrete analog of convexity, is a key property in discrete optimization that features in the construction of valid inequalities and analysis of the greedy algorithm. In this paper, we broaden the approximate submodularity literature, which so far has largely focused on variants of greedy algorithms and iterative approaches. We define metrics that quantify … Read more

Convexifying Multilinear Sets with Cardinality Constraints: Structural Properties, Nested Case and Extensions

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained version of it with upper and lower bounds on the number of nonzero variables. We call the set of … Read more

Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality

Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch- and-cut has proven to be a powerful extension … Read more

Chance-Constrained Bin Packing Problem with an Application to Operating Room Planning

We study the chance-constrained bin packing problem, with an application to hospital operating room planning. The bin packing problem allocates items of random size that follow a discrete distribution to a set of bins with limited capacity, while minimizing the total cost. The bin capacity constraints are satisfied with a given probability. We investigate a … Read more

Intersection disjunctions for reverse convex sets

We present a framework to obtain valid inequalities for optimization problems constrained by a reverse convex set, which is defined as the set of points in a polyhedron that lie outside a given open convex set. We are particularly interested in cases where the closure of the convex set is either non-polyhedral, or is defined … Read more

A New Extended Formulation with Valid Inequalities for the Capacitated Concentrator Location Problem

In this paper, we first present a new extended formulation of the Capacitated Concentrator Location Problem (CCLP) using the notion of cardinality of terminals assigned to a concentrator location. The disaggregated formulation consists of O(mn2) variables and constraints, where m denotes the number of concentrators and n the number of terminals. An immediate benefit of … Read more

Submodularity and valid inequalities in nonlinear optimization with indicator variables

We propose a new class of valid inequalities for mixed-integer nonlinear optimization problems with indicator variables. The inequalities are obtained by lifting polymatroid inequalities in the space of the 0–1 variables into conic inequalities in the original space of variables. The proposed inequalities are shown to describe the convex hull of the set under study … Read more

A multi-stage stochastic integer programming approach for a multi-echelon lot-sizing problem with returns and lost sales

We consider an uncapacitated multi-item multi-echelon lot-sizing problem within a remanufacturing system involving three production echelons: disassembly, refurbishing and reassembly. We seek to plan the production activities on this system over a multi-period horizon. We consider a stochastic environment, in which the input data of the optimization problem are subject to uncertainty. We propose a … Read more

Model and exact solution for a two-echelon inventory routing problem

The classic version of the Inventory Routing Problem considers a system with one supplier that manages the stock level of a set of customers. The supplier defines when and how much products to supply and how to combine customers in routes while minimizing storage and transportation costs. We present a new version of this problem … Read more

Lifted Polymatroid Inequalities for Mean-Risk Optimization with Indicator Variables

We investigate a mixed 0-1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalities … Read more