Study The Smoothing Method For Second-order Cone Complementarity

In this paper, we study the second-order cone complementarity problem (SOCCP for short). This problem is to find a vetor satisfying a system of equations and a complementarity condition defined on the Cartesian product of second-order cones, simultaneously, It has wide applications in engineering, economics, management science, and other fields. In the paper, firstly the theory, algorithm, and recent research of the second-cone complementarity problem is summarized, we propose three kinds of smoothing algorithm using Jordan algebraic technique. Then our some work in algorithm is introduced. For detail,we conclude them as follows:1. In this paper,based on the smoothing method for SOCCP,we propose a smoothing algorithm for solving the SOCCP with a non-monotone line search. This algorithm does not have restrictions regarding its starting point.We show that the algorithm is globally convergent and locally superlinearly convergent with a PO function, finally, the experiment is given and the data result prove that this algorithm is superior to the smoothing method.2. We extend the predictor-corrector smoothing method for LP to the second-order cone complementarity problem, Based on Chen and Mangasarian smoothing function, a non-interior-point predictor-corrector path following method is presented in this paper. This algorithm does not have restrictions regarding its starting point. We show that the algorithm is globally convergent and Q-quadratically convergent. finally,the experiment is given and the data result prove that this algorithm is superior to the predictor-corrector smoothing algorithm for SOCP.3. We extend the predictor-corrector smoothing method for LP to the second-order cone complementarity problem, Based on Chen and Mangasarian smoothing function, a predictor-corrector smoothing method for solving SOCCP is presented, the neighborhood of path does not appear in the algorithm, thus, it does not need a few additional computations which keep the iteration sequence staying in the given neighbourhood, the algorithm is simpler than the predictor-corrector smoothing algorithm for SOCP , it does not have restrictions regarding its starting point. We show that the algorithm is globally convergent and locally superlinearly convergent. finally, some preliminary computational results are reported.