Relaxation strength for multilinear optimization: McCormick strikes back

We consider linear relaxations for multilinear optimization problems. In a recent paper, Khajavirad proved that the extended flower relaxation is at least as strong as the relaxation of any recursive McCormick linearization (Operations Research Letters 51 (2023) 146-152). In this paper we extend the result to more general linearizations, and present a simpler proof. Moreover, … Read more

A proof system for certifying symmetry and optimality reasoning in integer programming

We present a proof system for establishing the correctness of results produced by optimization algorithms, with a focus on mixed-integer programming (MIP). Our system generalizes the seminal work of Bogaerts, Gocht, McCreesh, and Nordström (2022) for binary programs to handle any additional difficulties arising from unbounded and continuous variables, and covers a broad range of … Read more

Cutting planes from the simplex tableau for quadratically constrained optimization problems

We describe a method to generate cutting planes for quadratically constrained optimization problems. The method uses information from the simplex tableau of a linear relaxation of the problem in combination with McCormick estimators. The method is guaranteed to cut off a basic feasible solution of the linear relaxation that violates the quadratic constraints in the … Read more

Safe and Verified Gomory Mixed Integer Cuts in a Rational MIP Framework

This paper is concerned with the exact solution of mixed-integer programs (MIPs) over the rational numbers, i.e., without any roundoff errors and error tolerances. Here, one computational bottleneck that should be avoided whenever possible is to employ large-scale symbolic computations. Instead it is often possible to use safe directed rounding methods, e.g., to generate provably … Read more

On Supervalid Inequalities for Binary Interdiction Games

Supervalid inequalities are a specific type of constraints often used within the branch-and-cut framework to strengthen the linear relaxation of mixed-integer programs. These inequalities share the particular characteristic of potentially removing feasible integer solutions as long as they are already dominated by an incumbent solution. This paper focuses on supervalid inequalities for solving binary interdiction … Read more

Optimality-Based Discretization Methods for the Global Optimization of Nonconvex Semi-Infinite Programs

We use sensitivity analysis to design optimality-based discretization (cutting-plane) methods for the global optimization of nonconvex semi-infinite programs (SIPs). We begin by formulating the optimal discretization of SIPs as a max-min problem and propose variants that are more computationally tractable. We then use parametric sensitivity theory to design an efficient method for solving these max-min … Read more

Monoidal Strengthening of Simple V-Polyhedral Disjunctive Cuts

Disjunctive cutting planes can tighten a relaxation of a mixed-integer linear program. Traditionally, such cuts are obtained by solving a higher-dimensional linear program, whose additional variables cause the procedure to be computationally prohibitive. Adopting a V-polyhedral perspective is a practical alternative that enables the separation of disjunctive cuts via a linear program with only as … Read more

Integer Programming Models for Round Robin Tournaments

Round robin tournaments are omnipresent in sport competitions and beyond. We propose two new integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations with the linear relaxation of a well-known traditional formulation. We find that the matching formulation is stronger than … Read more

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations have been among the most popular convexification techniques for binary polynomial optimization problems. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal recursive sequence amounts to solving a difficult combinatorial optimization problem. In this paper, we prove that any … Read more

Relaxations and Cutting Planes for Linear Programs with Complementarity Constraints

We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and generalizes the extended reformulation-linearization technique of Nguyen, Richard, and Tawarmalani to instances with general complementarity conditions between variables. We demonstrate … Read more