Sparsity Constrained Minimization via Mathematical Programming with Equilibrium Constraints

TL;DR

This paper introduces a novel approach using Mathematical Programs with Equilibrium Constraints (MPECs) to efficiently solve sparsity constrained minimization problems, outperforming traditional iterative hard thresholding methods.

Contribution

It reformulates sparsity constrained problems as biconvex MPECs and develops exact penalty and alternating direction methods with proven convergence.

Findings

MPEC-based methods outperform state-of-the-art techniques

Effective in feature selection and signal processing tasks

Convergence properties are rigorously analyzed

Abstract

Sparsity constrained minimization captures a wide spectrum of applications in both machine learning and signal processing. This class of problems is difficult to solve since it is NP-hard and existing solutions are primarily based on Iterative Hard Thresholding (IHT). In this paper, we consider a class of continuous optimization techniques based on Mathematical Programs with Equilibrium Constraints (MPECs) to solve general sparsity constrained problems. Specifically, we reformulate the problem as an equivalent biconvex MPEC, which we can solve using an exact penalty method or an alternating direction method. We elaborate on the merits of both proposed methods and analyze their convergence properties. Finally, we demonstrate the effectiveness and versatility of our methods on several important problems, including feature selection, segmented regression, MRF optimization, trend filtering and impulse noise removal. Extensive experiments show that our MPEC-based methods outperform state-of-the-art techniques, especially those based on IHT.