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arXiv · 1407.7205

A Smoothing SQP Framework for a Class of Composite $L_q$ Minimization over Polyhedron

Ya-Feng Liu·Shiqian Ma·Yu-Hong Dai·Shuzhong Zhang
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Abstract

The composite $L_q~(0<q<1)$ minimization problem over a general polyhedron has received various applications in machine learning, wireless communications, image restoration, signal reconstruction, etc. This paper aims to provide a theoretical study on this problem. Firstly, we show that for any fixed $0<q<1$, finding the global minimizer of the problem, even its unconstrained counterpart, is strongly NP-hard. Secondly, we derive Karush-Kuhn-Tucker (KKT) optimality conditions for local minimizers of the problem. Thirdly, we propose a smoothing sequential quadratic programming framework for solving this problem. The framework requires a (approximate) solution of a convex quadratic program at each iteration. Finally, we analyze the worst-case iteration complexity of the framework for returning an $ε$-KKT point; i.e., a feasible point that satisfies a perturbed version of the derived KKT optimality conditions. To the best of our knowledge, the proposed framework is the first one with a worst-case iteration complexity guarantee for solving composite $L_q$ minimization over a general polyhedron.

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Ya-Feng Liu, Shiqian Ma, Yu-Hong Dai, Shuzhong Zhang. 2014-07-27. A Smoothing SQP Framework for a Class of Composite $L_q$ Minimization over Polyhedron. https://arxiv.org/abs/1407.7205

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