The composite Lq(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. First, we derive the Karush–Kuhn–Tucker (KKT) optimality conditions for local minimizers of the problem. Second, 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 Lq minimization over a general polyhedron.
Bibliographical noteFunding Information:
Y.-F. Liu was partially supported by NSFC Grants 11331012 and 11301516. S. Ma was partially supported by the Hong Kong Research Grants Council General Research Fund Early Career Scheme (Project ID: CUHK 439513). Y.-H. Dai was partially supported by the China National Funds for Distinguished Young Scientists Grant 11125107, the Key Project of Chinese National Programs for Fundamental Research and Development Grant 2015CB856000, NSFC Grant 71331001, and the CAS Program for Cross & Cooperative Team of the Science & Technology Innovation. S. Zhang was partially supported by NSF Grant CMMI-1462408.
- Composite L minimization
- Nonsmooth nonconvex non-Lipschitzian optimization
- Optimality condition
- Smoothing approximation
- Worst-case iteration complexity
- ϵ-KKT point