Study On The Algorithms For Symmetric Cone Complementarity Problem

Symmetric Cone Complementarity Problem(SCCP) is a class of important equilib-rium optimization problems with new content, abundant theory, and extensive applica-tion. The SCCP provides a simple uni?ed framework for various existing complementarityproblems such as nonlinear complementarity problem (NCP), second-order cone comple-mentarity problem (SOCCP), and the semide?nite complementarity problem (SDCP). Italso has close relation with the combination optimization, robust optimization, uncertainoptimization, and game and equilibrium theory.This thesis is devoted to study the smoothing Newton method for solving severalclass of SCCP, including the monotone SCCP, the special case of monotone SCCP andtwo classes of nonmonotone SCCP, and to investigate some properties of the merit functionof SCCP by the tool of Euclidean Jordan algebras. When the smoothing Newton methodis used to solve SCCP, we ?rst reformulate it into a nonlinear system of non-smooth equa-tions by the complementarity function, such as the minimum complementarity functionand the Fischer-Burmeister(FB) complementarity function. And then a smooth factor isintroduced into the complementary function. So the non-smooth reformulation equationsare smoothed, seeming the smooth factor as a variable. Finally, the Newton method isused to solve the nonlinear system of smooth equation. The main contributions are listedas follows:For the monotone SCCP, we proposed a predictor-corrector smoothing Newtonmethod based on the symmetrically perturbed smoothing function. Under a mild as-sumption that the solution set of the problem concerned is just nonempty, we prove theglobal convergence of the proposed algorithm. And the local superlinear convergence isobtained under the suitable assumption. Also, we extended a class of new smoothingfunction of second order cone complementarity function to the SCCP, and researched theproperties of the new smoothing function. Based on this new smoothing function, we pro-posed a one step smoothing Newton method for monotone SCCP. The well-de?nednessof the method and the global convergence and the local superlinear convergence wereobtained.For the monotone SOCCP, we presented a one step smoothing Newton methodbased on a class of parametric smoothing function. The well-de?nedness and the con- vergence were researched. We also give a numerical example about 2×2 P₀-matrix,which implies that the smoothing Newton method based on Chen-Harker-Kanzow-Smale(CHKS) smoothing function (when P = 0) can not be used for solving the class of non-monotone P₀-SOCCP. At last, the preliminary numerical results are also reported to showthe in?uence of the parametric to the numerical e?ect.For the nonmonotone Symmetric Cone Linear Complementary Problem(SCLCP)with the Cartesian P-property, we proposed a smoothing Newton method based on theCHKS smoothing function. We proved the nonsingularity of Jacobian matrices under thecondition of the Cartesian P₀-property and the boundedness of neighborhood of iteratesgenerated by the smoothing Newton method. Hence, the well-de?nedness and the globaland local quadratic convergence were obtained.For the nonmonotone SCLCP with the Cartesian P₀-property, based the CHKSsmoothing function we presented a regularization smoothing Newton method, and thewell-de?nedness and the convergence are analyzed. Based on the famous symmetric per-turbed Fischer-Burmeister smoothing function, a smoothing Newton method is proposed.We proved the nonsingularity of Jacobian matrices (which implies the solvability of New-ton's equation) and the coerciveness of the target function (which implies the boundednessof the neighborhood of iterates) under the condition of Cartesian P₀-property. Moreover,the global convergence is obtained under a nonsingularity assumption.Based on the Euclidean Jordan algebra, we proposed a new merit function forSCCP, studied the condition under which the level set of the merit function is boundedand the merit function provided a global error bound for the solution to the SCCP. Thetwo properties can be used to provide the stop criterion and to analyzed the convergenceof the algorithm. Also, for the existing merit function, a weaker condition under whichthe existing merit function have the above two properties was proposed.