The Applications Of Nonmonotone Technique And Filter Method In Optimization And Nonsmooth Equations

In this thesis,we mainly study the applications of nonmonotone technique and filter method in optimization and nonsmooth equations.Filter techniques have been applied successfully in smooth nonlinear optimization and nonlinear equations.We apply filter technique to nonsmooth optimization and nonsmooth equations,and combine filter technique,conic model and trust-region technology to present a conic trust-region filter method for nonlinearly constrained optimization problems.We present these algorithms and prove the convergence of these algorithms,and report some numerical results for some algorithms.In chapter 1,we introduce some notations and definitions that we use in this thesis,and briefly introduce some knowledge in convex analysis,nonsmooth analysis,nonmonotone technique and filter method.In chapter 2,we mainly study the filter-trust-region method for solving nonsmooth equations,where the function is locally Lipschitzian.The algorithm attempts to combine the efficiency of filter techniques and the robustness of trust-region method.The algorithm is also an extension of the classic Levenberg-Marquardt method by approximating the locally Lipschitzian function with a smooth function and using the derivative of the smooth part in the algorithm wherever a derivative is needed.Global convergence for this algorithm is established under reasonable assumptions.In chapter 3,we mainly study the filter-trust-region method for LC~1 unconstrained optimization problems,which use the second Dini upper directional derivative.This algorithm is the further extension of the filter techniques to the unconstrained optimization problems[31].The new algorithm is shown to be globally convergent under reasonable assumptions.In chapter 4,we mainly study the conic trust-region filter method for nonlinearly constrained optimization problems.Trust-region methods are powerful optimization methods.The conic model method is a new type of methods with more information available at each iteration than standard quadratic-based methods.Filter methods proposed by Fletcher and Leyffer,offering an alternative to merit functions,can guarantee global convergence of algorithms for nonlinear programming.In this chapter,we present a conic trust-region filter method for nonlinearly constrained optimization problems.The new algorithm is shown to be globally convergent under standard conditions.In chapter 5,we mainly study nonmonotonic trust region method for unconstrained optimization problem.In order to guarantee the global convergence of an algorithm for unconstrained optimization,the usual trust region methods force a monotonical decrease of the objective function at each step,which sometimes can considerably slow the rate of convergence.A nonmonotonic trust region method presented in this paper allows an increase in the function value at some step,while retaining global convergence and superlinear convergence.Finally, numerical experiments show that the nonmonotonic trust region method is superior to the usual trust region methods.