The Study Of The Second Order Cone Linear Complementarity Problems

The second-order cone complementarity problem(SOCLCP)is an important equilibrium optimization problem.It is to find a vector to satisfy a system of equations and the complementarity condition on the second-order Cartesian product under the condition of the second order cone constraint.The main idea is to get the optimal solution of complementary problem by using mathematical method,computer and network tools.The research of the second-order cone problem has obtained rich theoretical results and many kinds of methods were gave to solve it,such as the value function method,the smooth Newton method,the semi-smooth Newton method,the interior-point method,the power penalty method and matrix splitting method.The application fields involve finance,control,engineering technology,combinatorial optimization,neural network,machine learning and other fields.However,with the advent of big data age,the matrix involved in the second-order cone problem is largescale sparse matrix.So there are still lots of problems that need to be studied.In this thesis,we study the second-order cone linear complementarity problem.First,we summarize the basic knowledge of the theory,algorithm and research status of the second-order cone linear complementarity problem.Then,on the basis of transforming the problem into an equivalent fixed point equation,three Newton algorithms are designed for solving the problem,and the convergence is proved.At last,the corresponding numerical results show our algorithm is effective.This thesis is organized as below:In the first chapter,we introduce the research background of the second-order cone complementarity problem,the basic symbols,concepts and lemmas to prove the convergence of our algorithm.In the second chapter,based on the characteristics of the solution and the matrix equation,we proposed parameter-Newton method and the pre-conditioned parameterNewton method.The corresponding convergence analysis is given,which proves the algorithm can at least quadratic convergence to the exact solution.Finally,different numerical experiments also show the feasibility and effectiveness of the proposed method.In the third chapter,in order to avoid subcase discussion,we propose the squareNewton method by introducing two parameters for large-scale sparse matrix.Then the convergence analysis proves that the algorithm converges superlinearly to the exact solution.Finally,different numerical examples are used to verify effectiveness and superiority of our algorithm.In the last chapter,we summarize all the contents and algorithms,and put forward some ideas that can be studied in the future.