Learning to Relax Nonconvex Quadratically Constrained Quadratic Programs

Abstract: Quadratically constrained quadratic programs (QCQPs) are ubiquitous in optimization: Such problems arise in applications from operations research, power systems, signal processing, chemical engineering, portfolio theory, among others. Despite their flexibility in modeling real-life situations and the recent effort to understand their properties, nonconvex QCQPs are hard to solve in practice. Most of the approaches in the literature are based on either Linear Programming (LP) or Semidefinite Programming (SDP) relaxations, each of which works very well for some problem subclasses but perform poorly on others. In this paper, we develop a relaxation selection procedure for nonconvex QCQPs that can adaptively decide whether an LP- or SDP-based approach is expected to be more beneficial by considering the instance structure. The proposed methodology relies on utilizing machine learning methods that involve features derived from spectral properties and sparsity patterns of data matrices, and once trained appropriately, the prediction model is applicable to any instance with an arbitrary number of variables and constraints. We train and test classification and regression models over synthetically generated instances, and empirically show the efficacy of our approach.