Solving Graph Partitioning on Sparse Graphs: Cuts, Projections, and Extended Formulations

This paper explores new solution approaches for the graph partitioning problem. While the classic formulations for graph partitioning are compact, they either suffer from a poor relaxation, symmetry, or contain a cubic number of constraints regardless of the graph density. These shortcomings often result in poor branch-and-bound performance. We approach this problem from perspective of … Read more

Second-Order Conic and Polyhedral Approximations of the Exponential Cone: Application to Mixed-Integer Exponential Conic Programs

Exponents and logarithms exist in many important applications such as logistic regression, maximum likelihood, relative entropy and so on. Since the exponential cone can be viewed as the epigraph of perspective of the natural exponential function or the hypograph of perspective of the natural logarithm function, many mixed-integer nonlinear convex programs involving exponential or logarithm … Read more

Sequential Competitive Facility Location: Exact and Approximate Algorithms

We study a competitive facility location problem (CFLP), where two firms sequentially open new facilities within their budgets, in order to maximize their market shares of demand that follows a probabilistic choice model. This process is a Stackelberg game and admits a bilevel mixed-integer nonlinear program (MINLP) formulation. We derive an equivalent, single-level MINLP reformulation … Read more

A Survey on Mixed-Integer Programming Techniques in Bilevel Optimization

Bilevel optimization is a field of mathematical programming in which some variables are constrained to be the solution of another optimization problem. As a consequence, bilevel optimization is able to model hierarchical decision processes. This is appealing for modeling real-world problems, but it also makes the resulting optimization models hard to solve in theory and … 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

A Unified Framework for Multistage and Multilevel Mixed Integer Linear Optimization

We introduce a unified framework for the study of multilevel mixed integer linear optimization problems and multistage stochastic mixed integer linear optimization problems with recourse. The framework highlights the common mathematical structure of the two problems and allows for the development of a common algorithmic framework. Focusing on the two-stage case, we investigate, in particular, … Read more

Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

We study a family of discrete optimization problems asking for the maximization of the expected value of a concave, strictly increasing, and differentiable function composed with a set-union operator. The expected value is computed with respect to a set of coefficients taking values from a discrete set of scenarios. The function models the utility function … Read more

The SCIP Optimization Suite 7.0

The SCIP Optimization Suite provides a collection of software packages for mathematical optimization centered around the constraint integer programming framework SCIP. This paper discusses enhancements and extensions contained in version 7.0 of the SCIP Optimization Suite. The new version features the parallel presolving library PaPILO as a new addition to the suite. PaPILO 1.0 simplifies … Read more

Computational study of a branching algorithm for the maximum k-cut problem

This work considers the graph partitioning problem known as maximum k-cut. It focuses on investigating features of a branch-and-bound method to efficiently obtain global solutions. An exhaustive experimental study is carried out for two main components of a branch-and-bound algorithm: computing bounds and branching strategies. In particular, we propose the use of a variable neighborhood … Read more

Compact Formulations for Split Delivery Routing Problems

Split delivery routing problems are concerned with serving the demand of a set of customers with a fleet of capacitated vehicles at minimum cost, where a customer can be served by more than one vehicle if beneficial. They generalize traditional variants of routing problems and have applications in commercial as well as humanitarian logistics. Previously, … Read more