Quadratic Programming with Sparsity Constraints via Polynomial Roots

TL;DR

This paper introduces polynomial root-based approximations for solving sparse quadratically constrained quadratic programs, enabling more tractable solutions for problems like sparse linear regression and PCA.

Contribution

It presents a novel approach using polynomial roots derived from hyperbolic polynomials to approximate and solve sparse QCQPs efficiently.

Findings

Effective in practical problem instances

Outperforms some existing heuristics

Provides a new theoretical framework for sparse QCQPs

Abstract

Quadratically constrained quadratic programs (QCQPs) are an expressive family of optimization problems that occur naturally in many applications. It is often of interest to seek out sparse solutions, where many of the entries of the solution are zero. This paper will consider QCQPs with a single linear constraint, together with a sparsity constraint that requires that the set of nonzero entries of a solution be small. This problem class includes many fundamental problems of interest, such as sparse versions of linear regression and principal component analysis, which are both known to be very hard to approximate. We introduce a family of tractable approximations of such sparse QCQPs using the roots of polynomials which can be expressed as linear combinations of principal minors of a matrix. These polynomials arose naturally from the study of hyperbolic polynomials. Our main contributions are formulations of these approximations and computational methods for finding good solutions to a sparse QCQP. We will also give numerical evidence that these methods can be effective on practical problems.