On obtaining the convex hull of quadratic inequalities via aggregations

A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran showed … Read more

Cutting Plane Generation Through Sparse Principal Component Analysis

Quadratically-constrained quadratic programs (QCQPs) are optimization models whose remarkable expressiveness has made them a cornerstone of methodological research for nonconvex optimization problems. However, modern methods to solve a general QCQP fail to scale, encountering computational challenges even with just a few hundred variables. Specifically, a semidefinite programming (SDP) relaxation is typically employed, which provides strong … Read more

On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens … Read more

Outer-Product-Free Sets for Polynomial Optimization and Oracle-Based Cuts

Cutting planes are derived from specific problem structures, such as a single linear constraint from an integer program. This paper introduces cuts that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set $S\cap P$, where $S$ is a closed set, … Read more

LP formulations for mixed-integer polynomial optimization problems

We present polynomial-time algorithms for constrained optimization problems overwhere the intersection graph of the constraint set has bounded tree-width. In the case of binary variables we obtain exact, polynomial-size linear programming formulations for the problem. In the mixed-integer case with bounded variables we obtain polynomial-size linear programming representations that attain guaranteed optimality and feasibility bounds. … Read more