Nonsingularity Study Of The Parametric FB System For The Second-order Cone Programming

Second-order cone progrmamming is a class of nonpolyhedral convex conic opti-mization problems,which has important applications in engineering design，system con-trol,finance,robust optimization, and so on. In this paper,we study the strong regularityof local optimal solutions of the nonlinear second-order cone programming (SOCP) bythe nonsingylarity of Clarke’s generalized Jacobians of the FB complementarity system.Specifically,for a local optimal solution of the nonlinear second-order cone program-ming,under Robinson’s constraint qualification we estabilished the equivalence among thefollowing conditions: strong regularity of the KKT point,the strong second-order suf-cient condition and constraint nondegeneracy,the nonsingularity of Clark’s Jacobian ofthe smoothed FB system at the KKT point, and nonsingularity of Clark’s Jacobian ofthe parametric FB system at the KKT point,by studying the properties of directionalderivative and Clark’s Jacobian of the smoothed FB SOC complementarity function andthe parametric FB SOC complementarity function.In particular,for the linear SOCP, weobtained that equivalence among the primal and dual constraint nondegenercy, the non-singularity of Clark’s Jacobian of the smoothed FB system, the strong regularity of theKKT point.The thesis is organized as follows: Chapter1briefly reviews the existing efectivealgorithms for solving the nonlinear SOCPs and introduced the main contents of this the-sis. Chapter2introduces the preliminary knowledge and the lemmas that will be usedin the subsequent chapters. Chapter3establishs the strong regularity of local optimalsolutions of the nonlinear SOCP by studying the nonsingularity of Clarke’s Jacobians ofthe smoothed FB SOC complementarity system.Chapter4establishes the strong regu-larity of local optimal solutions of the nonlinear SOCP by studying the nonsingularity ofClarke’s Jacobians of the parametric FB SOC complementarity system.