The Theory And Numerical Methods For Symmetric Cone Complementarity Problems

As one of the most important equilibrium optimization problems,the symmetric cone com-plementarity problem has been widely used in the fields of economy,communication engi-neering,transportation and provides a unified framework for the nonlinear complementarity problem,the second-order cone complementarity problem and the semidefinite complemen-tarity problem.Moreover,it is bound up with the optimality conditions of the symmetric cone linear programming,combinatorial optimization,uncertain optimization and equilib-rium theory.Euclidean Jordan algebra is a powerful tool for the symmetric cone comple-mentarity problem.In virtue of Euclidean Jordan algebras,this dissertation studies several symmetric cone complementarity problems,including the monotone symmetric cone com-plementarity problem,the strongly monotone symmetric cone complementarity problem and the Cartesian P₀-symmetric cone complementarity problem and presents the corresponding algorithms.The main contributions are as follows.Firstly,a new projection and contraction method of the monotone symmetric cone comple-mentarity problem is presented.On the basis of the Frobenius norm and the projection oper-ator onto the symmetric cone,the symmetric cone complementarity problem is transformed into an equivalent projection equation.The solution of the symmetric cone complementar-ity problem can be obtained by means of solving the projection equation.The projection and contraction method only requires some projection calculations and functional computa-tions.It is proved that the iteration sequence produced by the proposed method converges to a solution of the symmetric cone complementarity problem under the condition that the underlying transformation is monotone.Numerical experiments also show the effectiveness of this method.Secondly,the projection method is further studied and a new projected Barzilai-Borwein method for the complementarity problem over symmetric cone is proposed by applying the Barzilai Borwein-like steplengths to the projection method.The projected Barzilai-Borwein method employs a new direction whose descent property depends on the fact that the trans-formation is strongly monotone and Lipschitz continuous.Meanwhile,a non-monotone line search is used in order to guarantee the global convergence.Some preliminary computa-tional results are also reported which verify the good theoretical properties of the proposed method.Thirdly,a regularization smoothing Newton method with a nonmonotone line search of the Cartesian P₀-symmetric cone linear complementarity problem is proposed.The global con-vergence of the new method does not require the solution set to be bounded which is dif-ferent from most smoothing Newton methods.It is shown that the sequence produced by the method is bounded only if the solution set of the complementarity problem is nonempty,and then the proposed method possesses global convergence.Some experimental results are reported to illustrate the efficiency of the proposed method.Finally,the Cartesian P₀-symmetric cone complementarity problem is considered and a new smoothing Newton method is proposed on the basis of a new class of smoothing function and a nonmonotone line search.The new class of smoothing function can be seen as a gen-eralized Fischer-Burmeister smoothing function.The smoothing Newton method is globally convergent when the problem has a nonempty and bounded solution set.It is proved that the smoothing Newton method is locally superlinearly or quadratically convergent under mild conditions.Preliminary numerical results are also reported which indicate the proposed method is promising.