Using K-T optimality condition of nonsmooth optimization, we establish two equivalent systems of the nonsmooth equations for the constrained minimax problem directly. Then generalized Newton methods are applied to so...Using K-T optimality condition of nonsmooth optimization, we establish two equivalent systems of the nonsmooth equations for the constrained minimax problem directly. Then generalized Newton methods are applied to solve these systems of the nonsmooth equations. Thus a new approach to solving the constrained minimax problem is developed.展开更多
A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk...A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA- LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.展开更多
Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equatio...Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equations are developed and their convergence is shown. Since this subdifferential is easy to be computed, the present Newton methods can be executed easily in some applications.展开更多
The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right...The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 < μ < 1) is a curve with the end points z = ?1, 1. L 1, L 2, L 3, L 4 are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y < 0}) passing through the points z = x + iy = ?1, 0, 0, 1 respectively. And the boundary condition possesses the form $$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H 1(y)u x ? iH 2(x)u y]/2 i展开更多
Presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. Details of the radius of the weak Jacobian of P...Presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. Details of the radius of the weak Jacobian of Picard iteration function; Generalized Jacobian; Generalized Newton methods for piecewise equations.展开更多
In this paper. we present a class of' embedding methods for nonsmooth equations. Under suitable conditions, we Prove that there exists a homotopy solution curve, which is Unique and continuous. We also prove that ...In this paper. we present a class of' embedding methods for nonsmooth equations. Under suitable conditions, we Prove that there exists a homotopy solution curve, which is Unique and continuous. We also prove that the solution curve is singlcvalue-d with respect to the homotopy parameter. Then we construct all efficient algorithm for this class of equations and prove its convcrgcnce. Filially, we apply the algorithm to the nonlinear complementarity problem. The numerical results show that tile algorithm is satisfacotry.展开更多
Newton-FOM （Full Orthogonalization Method ） algorithm and NewtonGMRES （Generalized Minimum Residual Method） algorithm for solving nonsmooth equations are presented. It is proved that these Krylov subspace algorith...Newton-FOM （Full Orthogonalization Method ） algorithm and NewtonGMRES （Generalized Minimum Residual Method） algorithm for solving nonsmooth equations are presented. It is proved that these Krylov subspace algorithms have the locally quadratic convergence. Numerical experiments demonstrate the effectiveness of the algorithms.展开更多
In this paper, we establish an inexact parameterized Newton method for solving the B differentiable equations. By introducing a new concept, we prove the local and large range convergence of the method under some wea...In this paper, we establish an inexact parameterized Newton method for solving the B differentiable equations. By introducing a new concept, we prove the local and large range convergence of the method under some weaker assumptions. We have conducted some numerical experiments. The numerical results show that the method is effective.展开更多
文摘Using K-T optimality condition of nonsmooth optimization, we establish two equivalent systems of the nonsmooth equations for the constrained minimax problem directly. Then generalized Newton methods are applied to solve these systems of the nonsmooth equations. Thus a new approach to solving the constrained minimax problem is developed.
基金Acknowledgments. This work is supported by the National Natural Science Foundation of China under projects Nos. 11071029, 11101064 and 91130007 and speciMized Research Fund for the Doctoral Program of Higher Education （20110041120039）. We are grateful to the associate editor and anonymous referee＇s comments to improve the quality of the manuscript. The second author also appreciate the discussion with his student Miao Xiaonan.
文摘A parameter-self-adjusting Levenberg-Marquardt method （PSA-LMM） is proposed for solving a nonlinear system of equations F（x） = 0, where F ：R^n→R^n is a semismooth mapping. At each iteration, the LM parameter μk is automatically adjusted based on the ratio between actual reduction and predicted reduction. The global convergence of PSA- LMM for solving semismooth equations is demonstrated. Under the BD-regular condition, we prove that PSA-LMM is locally superlinearly convergent for semismooth equations and locally quadratically convergent for strongly semismooth equations. Numerical results for solving nonlinear complementarity problems are presented.
文摘Numerical methods for the solution of nonsmooth equations are studied. A new subdifferential for a locally Lipschitzian function is proposed. Based on this subdifferential, Newton methods for solving nonsmooth equations are developed and their convergence is shown. Since this subdifferential is easy to be computed, the present Newton methods can be executed easily in some applications.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671207)
文摘The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 < μ < 1) is a curve with the end points z = ?1, 1. L 1, L 2, L 3, L 4 are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y < 0}) passing through the points z = x + iy = ?1, 0, 0, 1 respectively. And the boundary condition possesses the form $$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H 1(y)u x ? iH 2(x)u y]/2 i
基金The National Natural Science Foundation of China(11171221)the Research Fund for the Doctoral Program of Higher Education of China(20123120110004)+1 种基金the Natural Science Foundation of Shanghai(14ZR1429200)the Innovation Program of Shanghai Municipal Education Commission(15ZZ073)
基金The research was partly supported by NNSFC(No. 19771047) and NSF of Jiangsu Province (BK97059 ).
文摘Presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. Details of the radius of the weak Jacobian of Picard iteration function; Generalized Jacobian; Generalized Newton methods for piecewise equations.
文摘In this paper. we present a class of' embedding methods for nonsmooth equations. Under suitable conditions, we Prove that there exists a homotopy solution curve, which is Unique and continuous. We also prove that the solution curve is singlcvalue-d with respect to the homotopy parameter. Then we construct all efficient algorithm for this class of equations and prove its convcrgcnce. Filially, we apply the algorithm to the nonlinear complementarity problem. The numerical results show that tile algorithm is satisfacotry.
文摘Newton-FOM （Full Orthogonalization Method ） algorithm and NewtonGMRES （Generalized Minimum Residual Method） algorithm for solving nonsmooth equations are presented. It is proved that these Krylov subspace algorithms have the locally quadratic convergence. Numerical experiments demonstrate the effectiveness of the algorithms.
文摘In this paper, we establish an inexact parameterized Newton method for solving the B differentiable equations. By introducing a new concept, we prove the local and large range convergence of the method under some weaker assumptions. We have conducted some numerical experiments. The numerical results show that the method is effective.
