Global optimality condition and fixed point continuation algorithm for non-Lipschitz ?_p regularized matrix minimization  

作　　者：Dingtao Peng Naihua Xiu Jian Yu 

机构地区：[1]School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China [2]Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

出　　处：《Science China Mathematics》2018年第6期1139-1152,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.11401124 and 71271021);the Scientific Research Projects for the Introduced Talents of Guizhou University(Grant No.201343);the Key Program of National Natural Science Foundation of China(Grant No.11431002)

摘　　要：Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control,system identification and machine learning. In this paper, the non-Lipschitz ?_p(0 < p < 1) regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover,some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation(p-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz p （0 〈 p 〈 1） regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called p-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed p-thresholding fixed point continuation （p-FPC） algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.

关 键 词：lp regularized matrix minimization matrix completion problem p-thresholding operator globaloptimality condition fixed point continuation algorithm 

分 类 号：O224[理学—运筹学与控制论]