Research On A Lower Order Penalty Function Algorithm For Two Kinds Of Second-order Cone Linear Complementarity Problems

Second-order cone programming is affiliated to convex optimization problem in which the objective function is a linear function, it is a kind of problem which minimizes or maximizes a linear function over the intersection of affine space with the Cartesian product of a finite number of second-order cones. Linear programming, convex quadratic programming,quadratically constraint convex quadratic optimization as well as other problems can be translate into second-order cone Programming problems.They are a special case of semidefinite programming. Second-order cone programming has become a high-profile research direction in mathematical programming.Second-order cone complementarity problems refer to two decision variables satisfy a kind of complementary relationship with the second-order cone constraints, it is a class of equilibrium optimization problem. Second-order cone complementarity problems are affiliated to symmetric cone complementarity problems. In recent years,with the aid of the Euclidean Jordan algebra, the research of second-order cone Programming problems made many results, and made them absorb a lot of attention. Since the KKT conditions of secondorder cone programming can be transformed into second-order cone complementarity problems, so we can use the algorithm of second-order cone complementarity problems to solve second-order cone programming problems.The lower order penalty function algorithm is an effective method for solving symmetrical cone complementarity problems. The main think of lower order penalty function algorithm is convert the complementarity problems into lower order penalty equations, the prominent merit of this algorithm is that the solution sequence of the lower order penalty equations converges to the solution of the second-order cone complementarity problems at an exponential rate under a particular assumption. Since the lower order penalty function algorithm has many good properties, such as the accuracy of the solution an so on. Thus, it is worth to extend the lower order penalty function algorithm to solving second-order cone complementarity problems.The thesis primarily on the study of lower order penalty function algorithm for secondorder cone linear complementarity problems. The main contents are listed as follow.1. For second-order cone linear complementarity problems, based on the idea of lower order penalty function algorithm and the formula for the power of projection in second-order cone, it is converted to lower order penalty function equations. We prove that the solution sequence of the lower order penalty function equations converges to the solution of the secondorder cone linear complementarity problems at an exponential rate under definite matrix(not necessarily symmetrical) conditions. The results of numerical experiment demonstrate the effectiveness of the algorithm. We takes the numerical results of lower order penalty function algorithm to compare with the famous smooth Fischer-Burmeister(F-B) function algorithm,the results show that the proposed algorithm is effective.2. For a class of generalized second-order cone linear complementary problem, using the ideas of lower order penalty function algorithm, it is converted to lower order penalty function equations. Under the assumption that the matrix is positive definite(not necessarily symmetric), we prove that the solution sequence of the lower order penalty function equations converges to the solution of the generalized second-order cone complementarity problems at an exponential rate When the penalty parameter tends to infinite. The results of numerical experiment demonstrate the effectiveness of the algorithm.Finally, the main work of this paper are summarized and proposed further research work.