Sparse Solutions by a Quadratically Constrained lq (0 < q < 1) Minimization Model

Finding sparse solutions to a system of equations and/or inequalities is an important topic in many application areas such as signal processing, statistical regression and nonparametric modeling. Various continuous relaxation models have been proposed and widely studied to deal with the discrete nature of the underlying problem. In this paper, we propose a quadratically constrained l(q) (0< q < 1) minimization model for finding sparse solutions to a quadratic system. We prove that solving the proposed model is strongly NP-hard. To tackle the computation difficulty, a first order necessary condition for local minimizers is derived. Various properties of the proposed model are studied for designing an active-set-based descent algorithm to find candidate solutions satisfying the proposed condition. In addition to providing a theoretical convergence proof, we conduct extensive computational experiments using synthetic and real-life data to validate the effectiveness of the proposed algorithm and to show the superior capability in finding sparse solutions of the proposed model compared with other known models in the literature. We also extend our results to a quadratically constrained l(q) (0 < q < 1) minimization model with multiple convex quadratic constraints for further potential applications. Summary of Contribution: In this paper, we propose and study a quadratically constrained l(q) minimization (0 < q < 1) model for finding sparse solutions to a quadratic system which has wide applications in sparse signal recovery, image processing and machine learning. The proposed quadratically constrained l(q) minimization model extends the linearly constrained l(q) and unconstrained l(2)-l(q) models. We study various properties of the proposed model in aim of designing an efficient algorithm. Especially, we propose an unrelaxed KKT condition for local/global minimizers. Followed by the properties studied, an active-set based descent algorithm is then proposed with its convergence proof being given. Extensive numerical experiments with synthetic and real-life Sparco datasets are conducted to show that the proposed algorithm works very effectively and efficiently. Its sparse recovery capability is superior to that of other known models in the literature.