This paper is devoted to studying the generalized Jacobian for the projection onto the intersection of a closed half-space and a variable box. This paper derives the explicit formulas of an element in the set of the g...This paper is devoted to studying the generalized Jacobian for the projection onto the intersection of a closed half-space and a variable box. This paper derives the explicit formulas of an element in the set of the generalized HS Jacobian for the projection. In particular, we reveal that the generalized HS Jacobian can be formulated as the combination of a diagonal matrix and few rank-one symmetric matrices, which are crucial for future design of efficient second order nonsmooth methods for solving the related optimization problems.展开更多
We propose,in this paper,the preconditioned accelerated generalized successive overrelaxation(PAGSOR)iteration method for efficiently solving the large complex symmetric linear systems.To solve the nonlinear systems w...We propose,in this paper,the preconditioned accelerated generalized successive overrelaxation(PAGSOR)iteration method for efficiently solving the large complex symmetric linear systems.To solve the nonlinear systems whose Jacobian matrices are complex and symmetric,treating the PAGSOR method as internal iteration,we construct a modified Newton-PAGSOR(MN-PAGSOR)method to provide an effective approach for solving a wide range of problems in various scientific and engineering fields.Based on the Hölder continuous condition we present the theoretical framework of the modified method,demonstrate its local convergence properties,and provide numerical experiments to validate its effectiveness in solving a class of nonlinear systems.展开更多
Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of...Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.展开更多
文摘This paper is devoted to studying the generalized Jacobian for the projection onto the intersection of a closed half-space and a variable box. This paper derives the explicit formulas of an element in the set of the generalized HS Jacobian for the projection. In particular, we reveal that the generalized HS Jacobian can be formulated as the combination of a diagonal matrix and few rank-one symmetric matrices, which are crucial for future design of efficient second order nonsmooth methods for solving the related optimization problems.
基金National Natural Science Foundation of China with Grant Nos.12161030,12171216.
文摘We propose,in this paper,the preconditioned accelerated generalized successive overrelaxation(PAGSOR)iteration method for efficiently solving the large complex symmetric linear systems.To solve the nonlinear systems whose Jacobian matrices are complex and symmetric,treating the PAGSOR method as internal iteration,we construct a modified Newton-PAGSOR(MN-PAGSOR)method to provide an effective approach for solving a wide range of problems in various scientific and engineering fields.Based on the Hölder continuous condition we present the theoretical framework of the modified method,demonstrate its local convergence properties,and provide numerical experiments to validate its effectiveness in solving a class of nonlinear systems.
基金supported by National Natural Science Foundation of China(Grant Nos.10771026,10901094)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry of China
文摘Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker point.Moreover,this nonsingularity property guarantees the quadratic convergence of the corresponding smoothing Newton method for solving a Karush-Kuhn-Tucker point.Interestingly,the analysis does not need the strict complementarity condition.
