Nonsingularity Study Of The Penalized 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 [15,21]. In this paper, we use FB second-order cone complementarity function φρ and smoothing function φρ to transform KKT optimality conditions of Second-order cone programming problem into Nonsmooth system or smooth system to deal with. By studying directional derivative function and Clarke’s Jacobian of penalized FB Second-order cone complementarity function, under Robinson’s constraint qualification, we estabilished under the strong second-order sufficient condi-tion and constraint nondegeneracy, Clarke’s Jacobian of nonsmooth system Eρ(·)=0and smooth system Oρ(·)=0in KKT is nonsingular. By the modification method of equa-tions, We provide the smoothing complementarity function of φρ, and make numerical experiments base on smoothing Newton method for penalized FB smoothing complemen-tarity functionThe paper is organized as follows:Chapter one is introduction, we introduce the ex-isting algorithms of the nonlinear SOCPs, especially for the semismooth Newton method and smooth Newton method, also introduce the main contents of this paper. Chapter two gives the preliminary knowledge and the lemmas those will be used in the subsequent chapters. Chapter three, we study the properties of directional derivative function and Clarke’s Jacobian of penalized FB Second-order cone complementarity function, under the strong second-order sufficient condition to establish Clarke’s Jacobian of nonsmooth system Ep(·)=0and smooth system Oρ(·)=0in KKT is nonsingular. Chapter four, we prove a smoothing algorithm for the second-order cone program problem, We give numer-ical experiments from DIMACS collection standard linear second-order cone optimization problem, illustrate the smooth algorithm is superlinear convergence.