# An active-set method for sparse approximations. Part II: General piecewise-linear terms

## TL;DR

This paper introduces an efficient active-set method tailored for convex quadratic programming problems with general piecewise-linear terms, significantly improving memory efficiency and robustness for applications like risk minimization and sparse approximation.

## Contribution

It develops a novel active-set algorithm that exploits piecewise-linear structure to reduce memory use and enhance computational efficiency in convex quadratic programming.

## Key findings

- Successfully applied to risk-averse portfolio selection
- Effective in quantile regression and SVM classification
- Outperforms general-purpose solvers on real datasets

## Abstract

In this paper we present an efficient active-set method for the solution of convex quadratic programming problems with general piecewise-linear terms in the objective, with applications to sparse approximations and risk-minimization. The method exploits the structure of the piecewise-linear terms appearing in the objective in order to significantly reduce its memory requirements, and thus improve its efficiency. We showcase the robustness of the proposed solver on a variety of problems arising in risk-averse portfolio selection, quantile regression, and binary classification via linear support vector machines. We provide computational evidence to demonstrate, on real-world datasets, the ability of the solver of efficiently handling a variety of problems, by comparing it against an efficient general-purpose interior point solver as well as a state-of-the-art alternating direction method of multipliers. This work complements the accompanying paper [``An active-set method for sparse approximations. Part I: Separable $\ell_1$ terms", S. Pougkakiotis, J. Gondzio, D. S. Kalogerias], in which we discuss the case of separable $\ell_1$ terms, analyze the convergence, and propose general-purpose preconditioning strategies for the solution of its associated linear systems.