Enhancements of Discretization Approaches for Non-Convex Mixed-Integer Quadratically Constraint Quadratic Programming

We study mixed-integer programming (MIP) relaxation techniques for the solution of non-convex mixed-integer quadratically constrained quadratic programs (MIQCQPs). We present two MIP relaxation methods for non-convex continuous variable products that enhance existing approaches. One is based on a separable reformulation, while the other extends the well-known MIP relaxation normalized multiparametric disaggregation technique (NMDT). In addition, … Read more

Compact mixed-integer programming relaxations in quadratic optimization

We present a technique for producing valid dual bounds for nonconvex quadratic optimization problems. The approach leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky, formulating this (simple) approximation using mixed-integer programming (MIP). Notably, the number of constraints, binary variables, and auxiliary continuous variables used in this formulation grows logarithmically in … Read more

Complexity, Exactness, and Rationality in Polynomial Optimization

We study representation of solutions and certificates to quadratic and cubic optimization problems. Specifically, we focus on minimizing a cubic function over a polyhedron and also minimizing a linear function over quadratic constraints. We show when there exist rational feasible solutions (and if we can detect them) and when we can prove feasibility of sublevel … Read more

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The … Read more

Binary Extended Formulations of Polyhedral Mixed-integer Sets

We analyze different ways of constructing binary extended formulations of polyhedral mixed-integer sets with bounded integer variables and compare their relative strength with respect to split cuts. We show that among all binary extended formulations where each bounded integer variable is represented by a distinct collection of binary variables, what we call “unimodular” extended formulations … Read more

Extension Complexity Lower Bounds for Mixed-Integer Extended Formulations

We prove that any mixed-integer linear extended formulation for the matching polytope of the complete graph on $n$ vertices, with a polynomial number of constraints, requires $\Omega(\sqrt{\sfrac{n}{\log n}})$ many integer variables. By known reductions, this result extends to the traveling salesman polytope. This lower bound has various implications regarding the existence of small mixed-integer mathematical … Read more

Light on the Infinite Group Relaxation

This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled “Some continuous functions related to corner polyhedra I, II” [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context … Read more

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial in two variables over the integer points in a polyhedron. Previous work shows that in two variables, optimizing a quadratic polynomial over the integer points in a polyhedral region can be done in polynomial time, while optimizing a quartic polynomial in the same type … Read more

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. III. Foundations for the k-Dimensional Case with Applications to k=2

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, (1) we present the general regular solution to Cauchy’s additive functional equation on bounded convex domains. This provides a k-dimensional generalization of the so-called interval lemma, allowing us to deduce affine properties of the function from certain … Read more

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. II. The Unimodular Two-Dimensional Case

We give an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem. Article Download View Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. II. The Unimodular Two-Dimensional Case