An Alternating Bundle Method For A Class Of Nonsmooth Composite Optimizations

For nonsmooth composite optimization problems,minimizing the sum of the two functions is an important topic.It is widely used in realistic problems such as signal processing?multi-commodity flow?optimal control,and so on.They can be transformed into the optimization problem of the sum of two functions.Therefore,many researchers focus on the topic how to build feasible and efficient methods to solve them.The thesis focuses on a class of nonsmooth composite optimization problems,including minimizing the sum of two convex functions constrained by convex sets;minimizing the sum of a nonconvex function and a convex function and equilibrium problems of the sum of two nonsmooth bifunctions.We propose alternating numerical methods to solve nonsmooth optimization problems.The main results of this thesis can be summarized as follows:1.Chapter 3 concentrates on the interior alternating linearization bundle method for minimizing nonsmooth composite optimization problems which is the sum of two convex functions subject to a convex set.The main idea is as follows.For the proximal term of the algorithm,we use a second-order homogeneous distance-like function to replace the Euclidean norm.This assures that the iterations generated by the algorithm belong to the interior of the constraint set.Namely the algorithm will automatically generate feasible iterations.This is the main reason that the name "interior algorithms".The main steps are that we use an alternating linearization bundle method to solve two simple strongly convex subproblems.We obtain the convergence of the algorithm by the properties of second-order homogeneous distance-like functions.2.Chapter 4 is devoted to an alternating linearization bundle method for minimizing non-convex nonsmooth composite optimization problems,which is the sum of a a convex function and lower-C1 function with inexact data.The main idea is as follows.Firstly,we obtain locally convexification functions by the redistribution technique.Secondly,we use an bundle method to solve alternately two strongly convex quadratic programs.So we obtain the nonnegativity of the linearization errors.The global convergence of the algorithm is obtained under some weaker conditions.Finally,numerical results show that the alternating linearization bundle method is effective.3.Chapter 5 is devoted to a general nonsmooth equilibrium problem with a sum of two bifunctions,and presents an accelerated alternating method based on bundle techniques.The main idea is that we solve alternately two simple strongly convex subproblems at each iteration,combining an inertial method.Moreover for each nonsmooth subproblem,we provide an interior doubly bundle method to solve it.Without any Lipschitz continuous condition or H(?)lder continuity of the involved bifunctions,the convergence of the algorithm is proved.Finally,several experiments are performed to show the computational efficiency and provide comparisons with related algorithms.