An Proximal Alternating Bundle Method For A Class Of Nonconvex Nonsmooth Optimization Problems

Nonsmooth optimization is widely used in image denoising,neural net-work training,economics,and computational chemistry and physics,which can to be divided the convex nonsmooth optimization and nonconvex non-smooth optimization by the convexity of objective function and constrained function.In this thesis,we consider a class of nonconvex nonsmooth uncon-strained optimization problems.The objective function has a special struc-ture,which is the sum of a nonconvex function and a convex function.These problems are applied in a wide range,such as image reconstraction process-ing,compressed sensing,optimal control,system identification and so on.Therefore,it has theoretical and practical value to study the algorithm of solving these problems.In this thesis,we propose an alternating linearization bundle method for solving nonconvex nonsmooth optimization problems,which is the sum of a nonconvex function and a convex function.Firstly,we use locally con-vexification technique to locally convexificate the nonconvex function,that is,adding a quadratic term to the nonconvex function of objective function;Secondly,we construct a cutting plane model of the local convexification function to approximate the nonconvex function;Finally,we use alternating linearization method to obtain two simple subproblems by alternatively lin-earizing the cutting plane model and convex function.In the design of the algorithm,at each iteration,we just need to solve the two simple subprob-lems.This method extends the traditional alternating linearization bundle method from the convex to the nonconvex case.Furthermore,the new ad-justment strategy of proximal parameters and convexification parameters is designed to ensure the global convergence of the algorithm.At the end of this thesis,we conduct the numerical tests,and the numer-ical results show the feasibility,effectiveness and stability of the algorithm.