Quasi-newton Type Bundle Method For Nonconvex Nonsmooth Optimization

In this thesis,nonsmooth optimization(nondifferentiable optimization)problem is investigated,and objective function is not necessarily convex function,many practical problems can be summarized as such kind of prob-lems.Therefore,researching on robust and efficient numerical optimization methods for solving nonconvex nonsmooth optimization has important theo-retical significance and practical value.This thesis proposes a quasi-Newton type bundle method for solving nonconvex nonsmooth optimization,based on proximal bundle method and the idea of the quasi-Newton method,together with local convexification technic and Armijo line search rule.At each iteration,local convexifica-tion parameter ?e is modified by suitable strategy to overcome linearization error can be negative which due to the objective function without convexity assumption.Furthermore,if the candidate point satisfies the descent condi-tion,then it can be recognized as approximate proximal point of objective function at current prox-center.Based on approximate proximal point,we construct the approximate subgradient and approximate quasi-Newton direc-tion as line search direction.By monitoring the reduction in Euclidean norm of approximate subgradient to identify which way be used to determine step size,either take one as step size directly or make use of Armijo line search rule to compute step size.Under some mild conditions,global convergence of the proposed method is established,and the rate of convergence(linear convergence,superlinear convergence)is also discussed.Finally,with the help of the mathematical software MATLAB program-ming to verify the effectiveness and stability of the algorithm which is pro-posed in the thesis.Preliminary numerical results illustrate that the method is effective and robust.