The Theorey And Unified Algorithm For Minimization Of Locally Lipschitzian Functions

In this thesis, we study the non-smooth optimization problem where the objective function and constrained functions are all locally Lipschitzian functions. The thesis includes the generalized invexity of locally Lipschitzian functions; the generalized invariant monotonicity of set-value maps and the relationships between the generalized invariant monotonicity of convexificators and the generalized invexity of nonsmooth functions; the optimality necessary and sufficient conditions for non-linear programmings without a constraint qualification; mixed dual theory; Lagrange saddle point theory; nonmonotone line search methods for nonsmooth unconstrained optimization; nonmonotone trust region methods for nonsmooth unconstrained optimization.Using convexificators and invex functions introduced by Jeyakumar et al. and Hanson, respectively, the generalized convexity of nonsmooth functions and generalized monotonicity of set-valued maps are further extended. The relationships between the generalized invariant monotonicity of convexificators and the generalized invexity of nonsmooth functions are established.Without any constraint qualification, we establish the first-order optimality necessary and sufficient conditions for non-smooth scale programming, multi-objective programming and minmax fractinal programming involving pseudoin-vex functions, which generalizes the existing results. On the other hand, without the need of a constraint qualification, we establish the necessary and sufficient optimality conditions for generalized fractional programming involving a compact set. Using these optimality conditions, we construct a parametric dual model and a parameter-free mixed dual model, this mixed dual model unifies two dual parameter-free models constructed by Lai and Liu et al.. Several duality theorems are established.The nonmonotone line search methods have been successfully used in smooth unconstrained optimization and constrained optimization. Combining the forcing functions with the nonmonotone line search technique, we present a general framework of nonmonotone line search methods for the nonsmooth unconstrained optimization problems where the objective function is a locally Lipschitzian func-tion and establish the global convergence results . As special cases, we can obtain the nonsmooth nonmonotone Armijo rule, Goldstein rule and Wolfe rule.Combining the trust region algorithm of Qi and Sun with the nonmonotone technique, we present a nonmonotone trust region algorithm for the unconstrained nonsmooth optimization problems where the objective function is locally Lipschitzian, and prove that our nonmonotone trust region algorithm is globally convergent.