The Research Of Smoothing Function For Second-order Cone Complementarity Problems

The second order cone complementarity problems(SOCCP)are to find a vector under the condition of the second order cone constraint,make it satisfy a system of equations and meet the complementarity condition on the second-order Cartesian product.Its core idea is to get the optimal solution of complementary problem by using mathematical method and combining computer and network tools.The second-order cone complementarity problems with the theoretical background of finance,physics and computer programming language has attracted the atten-tion of various disciplines,especially the interdisciplinary field of mathematics and finance.Nowadays,the theory of complementary optimization has became an irre-placeable branch of mathematics.As the most basic and important component of the second-order cone complementarity problems,the research and promotion of it can enrich and improve the complementary optimization theory.This work first briefly introduces the basic knowledge of the theory,algorithm,and research status of the second-order cone complementarity problems.Secondly,with the help of smooth-ing ideas and based on commonly used second-order cone complementary functions,three new second-order cone complementary functions are given,and the properties of smooth functions are discussed.Finally,based on three new smooth complemen-tary functions,three Newton-like smoothing algorithms for solving second-order cone complementarity problems are designed and numerical results are given.The main research work is as follows:1.Based on the Fischer-Burmeister function combined with the positive vector function,a new second-order cone complementary function is given.An effective algorithm for solving the second-order cone complementarity problems are designed based on this function.Which proves the global convergence of the algorithm and give numerical experimental results.2.Based on the one-parameter smooth function of Liu and Huang,a new second-order cone complementary function is given in the framework of the CHKS function.This function contains two regularized forms of CHKS.It has good properties and is widely used.An effective algorithm for solving the second-order cone complementar-ity problems is given,and the wellness and global convergence of the algorithm are proved.Numerical experiments show that the function has good properties.3.Based on the second-order cone correlation theory,the weight complemen-tary function is degraded into a second-order cone smooth complementary function.Based on this degenerate smooth complementary function,a smooth Newton algorith-m based on weight complementary framework to solve the second-order cone com-plementarity problems.Prove the convergence of the algorithm under appropriate conditions.It can be seen from the comparison of numerical results that this degrad-ed smooth function can well solve the second-order cone complementarity problems.