The if-then Polytope: Conditional Relations over Multiple Sets of Binary Variables

Inspired by its occurrence as a substructure in a stochastic railway timetabling model, we study in this work a special case of the bipartite boolean quadric polytope. It models conditional relations across three sets of binary variables, where selections within two “if” sets imply a choice in a corresponding “then” set. We call this polytope … Read more

Using Disjunctive Cuts in a Branch-and-Cut Method to Solve Convex Integer Nonlinear Bilevel Problems

We present a branch-and-cut method for solving convex integer nonlinear bilevel problems, i.e., bilevel models with nonlinear but convex objective functions and constraints in both the upper and the lower level. To this end, we generalize the idea of using disjunctive cuts to cut off integer-feasible but bilevel-infeasible points. These cuts can be obtained by … Read more

Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8.0

For over ten years, the constraint integer programming framework SCIP has been extended by capabilities for the solution of convex and nonconvex mixed-integer nonlinear programs (MINLPs). With the recently published version 8.0, these capabilities have been largely reworked and extended. This paper discusses the motivations for recent changes and provides an overview of features that … Read more

Benders-type Branch-and-Cut Algorithms for Capacitated Facility Location with Single-Sourcing

We consider the capacitated facility location problem with (partial) single-sourcing (CFLP-SS). A natural mixed integer formulation for the problem involves 0-1 variables x_j indicating whether faclility j is used or not and y_{ij} variables indicating the fraction of the demand of client i that is satisfied from facility j. When the x variables are fixed, … Read more

Efficient MIP Techniques for Computing the Relaxation Complexity

The relaxation complexity rc(X) of the set of integer points X contained in a polyhedron is the minimal number of inequalities needed to formulate a linear optimization problem over X without using auxiliary variables. Besides its relevance in integer programming, this concept has interpretations in aspects of social choice, symmetric cryptanalysis, and machine learning. We … Read more

A solver for multiobjective mixed-integer convex and nonconvex optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are … Read more

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of … Read more

Hub Network Design Problem with Capacity, Congestion and Stochastic Demand Considerations

We introduce the hub network design problem with congestion, capacity, and stochastic demand considerations (HNDC), which generalizes the classical hub location problem in several directions. In particular, we extend state-of-the-art by integrating capacity acquisition decision and congestion cost effect into the problem and allowing dynamic routing for origin-destination pairs. Connecting strategic and operational level decisions, … Read more

The SCIP Optimization Suite 8.0

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 8.0 of the SCIP Optimization Suite. Major updates in SCIP include improvements in symmetry handling and decomposition algorithms, new cutting planes, a new plugin type … Read more