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

TL;DR

This paper studies the complex composite $L_q$ minimization problem over polyhedra, establishing NP-hardness, deriving optimality conditions, and proposing a smoothing SQP framework with iteration complexity guarantees.

Contribution

It introduces the first smoothing SQP framework with worst-case iteration complexity analysis for solving composite $L_q$ minimization over polyhedra.

Findings

NP-hardness of the problem for any fixed $0<q<1$

Derivation of KKT optimality conditions for local minimizers

Development of a smoothing SQP algorithm with complexity guarantees

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 $\epsilon$-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.