Effective Algorithms For Complementarity Problems Based On Nonsmooth Analysis

Nonsmooth optimization,which is an important content of technology level in system science,is widly used in the image denoising,optimal control,data mining and so on.In nonsmooth optimization,the traditional concept of differential is no longer applicable,a class of generalized differential form,such as Clarke subdifferential,B differential,quasi differential and so on,become an important part of nonsmooth theory.Considering the generalized differential of a general Lipchitzian function is not easy to calculate,we investigate the calculation of the generalized Jacobi of some special vector functions in this thesis,based on which some effective algorithms are proposed for the vertical complementarity problem,the mixed complementarity problem,the nonlinear complementarity problem,the nonlinear nonsmooth complementarity problem and so on.First of all,the vertical complementary problem,which is a kind of generalized complementarity problem,is inveatigated.A nonsmooth Levenberg-Marquardt algorithm(I)is proposed based on a new class of differential for the vertical complementary problem,which is with a self-adaptive LM parameter adjustment and the global convergence.Then,the computation of the generalized differential of a max-valued function is studied,based on which the nonsmooth Levenberg-Marquardt algorithm(II)is presented.Under local error bound,the local convergence rate is given.It is important to note that the non-smooth LM algorithm(II)modified the parameter adjustment scheme in the LM algorithm(I)theoretically.Secondly,the box-constrained mixed complementary problem is investigate.The mixed complementarity problem is transformed into two different systems of nonsmooth equations.By investigating the computation of the generalized differential of a kind of minimum composite function,a Levenberg-Marquardt algorithm for the mixed complementarity problem.Finally,the numerical experiments illustrate the effectiveness of the given method.As is well known,Jacobian smoothing method is a popular one to solve nonlinear complementarity problems,in which the degree of the first-order approximation between the constructed(or existed)smooth function and its corresponding nonsmooth function is often measured by the Jacobi consistency.In order to get smooth function better properties,the concept of strong Jacobi consistency is proposed,and the Jacobi the smoothing algorithm is also presented,based on the expression of an element in the related function's Clarke generalized Jacobi.Nonsmooth nonlinear complementarity problem is a kind of wide complementarity problem,in which the Clarke generalized Jacobi is difficult to obtain,a new kind of differential form is proposed in this thesis,based on which the local convergence rate of a Levenberg-Marquardt algorithm under the regular conditions is discussed.It is worth mentioning that the nonsmooth nonlinear complementarity problem contains other complemetarity problems,such as the complementary functions are convex,the complementarity functions are continuously differentiable and so on.