Nonsingularity in second-order cone programming via the smoothing metric projector  

作　　者：WANG Yun 1,& ZHANG LiWei 2 1 College of Information Sciences and Engineering,Shandong Agricultural University,Tai’an 271018,China 2 Department of Applied Mathematics,Dalian University of Technology,Dalian 116024,China 

出　　处：《Science China Mathematics》2010年第4期1025-1038,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos.10771026,10901094);the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry of China

摘　　要：Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke’s generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.

关 键 词：second-order cone programming problem SMOOTHING METRIC PROJECTOR B-subdifferential Clarke’s generalized JACOBIAN SMOOTHING Newton method 

分 类 号：O221.2[理学—运筹学与控制论]