# An active-set algorithm for norm constrained quadratic problems

## TL;DR

This paper introduces an active-set algorithm for solving nonconvex quadratic problems with norm constraints, leveraging efficient trust-region subproblem solutions and eigenpair computations, demonstrating superior scalability and performance.

## Contribution

It extends trust-region subproblem solutions to nonconvex quadratic problems with norm constraints, including efficient computation of nonglobal minimizers.

## Key findings

- Algorithm outperforms state-of-the-art methods in experiments.
- Effective in high-dimensional problems like Sparse PCA with 70 million nonzeros.
- Scalable approach applicable to large-scale optimization tasks.

## Abstract

We present an algorithm for the minimization of a nonconvex quadratic function subject to linear inequality constraints and a two-sided bound on the 2-norm of its solution. The algorithm minimizes the objective using an active-set method by solving a series of Trust-Region Subproblems (TRS). Underpinning the efficiency of this approach is that the global solution of the TRS has been widely studied in the literature, resulting in remarkably efficient algorithms and software. We extend these results by proving that nonglobal minimizers of the TRS, or a certificate of their absence, can also be calculated efficiently by computing the two rightmost eigenpairs of an eigenproblem. We demonstrate the usefulness and scalability of the algorithm in a series of experiments that often outperform state-of-the-art approaches; these include calculation of high-quality search directions arising in Sequential Quadratic Programming on problems of the CUTEst collection, and Sparse Principal Component Analysis on a large text corpus problem (70 million nonzeros) that can help organize documents in a user interpretable way.