Learning to Relax Nonconvex Quadratically Constrained Quadratic Programs

TL;DR

This paper introduces a machine learning-based method to adaptively select between LP and SDP relaxations for solving nonconvex QCQPs, improving solution efficiency across diverse problem instances.

Contribution

It develops a spectral and sparsity-based feature-driven classification/regression model that predicts the most effective relaxation method for any given QCQP instance.

Findings

The approach accurately predicts the better relaxation method for synthetic and benchmark instances.

Machine learning models outperform traditional heuristic selection methods.

The methodology is dimension-independent and adaptable to various QCQP structures.

Abstract

Quadratically constrained quadratic programs (QCQPs) are ubiquitous in optimization: Such problems arise in applications from operations research, power systems, signal processing, chemical engineering, and 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 applies to any instance with an arbitrary number of variables and constraints. We develop classification and regression models under different feature-design setups, including a dimension-independent representation, and evaluate them on both synthetically generated instances and benchmark instances from MINLPLib. Our computational results demonstrate the effectiveness of the proposed approach for predicting the more favorable relaxation across diverse QCQP families.