Research On Interior Point Method For Symmetric Cone Nonlinear Complementarity Problem

Symmetric cone complementarity problem is a very wide class of problems,including lin-ear and nonlinear complementarity problems,semi-definite complementarity problems and second order cone complementarity problems as special cases.It is widely used in econ-omy,management,traffic,engineer design,communication and control.In recent years,symmetric cone complementarity problem has become a very active research field,attract-ing a large number of scientists to engage in the research,and has achieved fruitful research results in theory,algorithms and applications.Interior point method(IPM)is one of the most effective may for solving symmetric cone complementarity problem.However,the research on the interior point algorithm for symmetric cone complementarity problem is mainly fo-cused on the symmetric cone linear complementarity problem.But there are few literatures about IPMs for symmetric cone nonlinear complementarity problems(SCNCPs).For mono-tone symmetric cone nonlinear complementarity problems and Cartesian P_*(?)symmetric cone nonlinear complementarity problems,this paper studies on homogenous algorithms,path-following interior point algorithm,and Mehrotra-type predictor-corrector algorithm,respectively.And estimates the complexity of interior point algorithm.Firstly,based on the homogenous algorithms proposed by Yoshise,this paper studies the ho-mogenous algorithms for symmetric cone monotone nonlinear complementarity problems.The proof of the complexity bounds requires that the nonlinear transformation satisfies a SLC.However,the SLC proposed by Yoshise depends on the scaling parameter p and hasn't scaled invariance.To prove the complexity bounds of the homogenous algorithm,this paper extends the condition proposed by Andersen et.al.from R_+~nto symmetric cone K and pro-poses a new SLC which does not depend on the scaling parameter p and is very easy to be verified.More important,it has scaled invariance.Underlying the SLC condition,the ob-tained complexity bounds of the short-step algorithm,the semi-long-step algorithm,and the long-step algorithm with sx direction match that of the homogenous algorithms proposed by Yoshise.Secondly,this paper studies a path-following interior point algorithm for Cartesian P_*(?)symmetric cone nonlinear complementarity problems(SCNCPs).This algorithm is an in-feasible algorithm based on a F-norm wide neighborhood.To estimate the theoretical com-plexity of the proposed algorithm,this paper proposes a SLC condition which has scaling invariance.Under the SLC condition,the iteration complexities of the proposed algorithm are estimated.Some numerical results of linear complementarity problems,semi-definite complementarity problems,and nonlinear complementarity problems are provided.The nu-merical results show that the algorithm is efficient and reliable.Finally.This paper studies two Mehrotra-type predictor-corrector algorithms for Carte-sian P_*(?)symmetric cone nonlinear complementarity problems.These two algorithms are Mehrotra-type predictor-corrector algorithms based on wide neighborhood N_?~-(1-?).The first algorithm extends the existing feasible predictor-correction algorithm for linear pro-gramming to the symmetric cone nonlinear complementarity problem.Different from the original algorithm,the extended algorithm is not feasible.At the same time,the adjustment strategy of central parameter which is different from the previous one is adopted.This paper proposes a theoretical framework of infeasible Mehrotra-type predictor-corrector algorith-m for Cartesian P_*(?)symmetric cone nonlinear complementarity problems.The iteration complexity of the algorithm is estimated and some numerical results of linear complemen-tarity problems,semi-definite complementarity problems,and nonlinear complementarity problems are provided.The numerical results show that the algorithm is efficient and re-liable.The second algorithm extends the Mehrotra-type predictor-corrector algorithm for Cartesian P_*(?)symmetric cone linear complementarity problems to Cartesian P_*(?)sym-metric cone nonlinear complementarity problems and establishes a theoretical framework of infeasible Mehrotra-type predictor-corrector algorithm for Cartesian P_*(?)symmetric cone nonlinear complementarity problems.The adjustment strategy of central parameter of the algorithm is different from that of the first algorithm,which can make the algorithm obtain a larger step size in each iteration and the theoretical complexity of the algorithm is estimated.The numerical results show that the proposed algorithm is efficient and reliable.