Proximal Bundle Methods For A Class Of Nonconvex And Nonsmooth Composite Optimization Problems

For the nonsmooth composite optimization problems,minimizing the sum of the two functions is an important topic.It is widely used in realistic problems such as Lasso problem in image reconstruction and optimization problems in machine learning and so on;it can also be transferred form splitting problems and nonlinear programming etc.When the problems own some special characteristics,some methods such as the splitting type methods and alternating direction type methods have nice numerical performance.However,the above methods may not be suitable if the problems are much hard to compute or the characteristics disappear.So it is meaningful to research new methods for the problems without the characters.The thesis focuses on a class of nonsmooth composite optimization problems and the main results of this thesis can be summarized as follows:1.In Chapter 2,a proximal composite bundle method is proposed for a class of nonconvex nonsmooth composite optimization problems.The composite problem considered here is the sum of two functions:one is convex and the other is nonconvex.In order to keep the linearization errors of the nonconvex function to be nonnegative,the convexification strategy is adopted and the corresponding convexification parameter varies along iterations.Then the sum of the convex function and the extended function is dynamically constructed to approximate the primal composite function.To choose a suitable cutting plane model for the approximation function,here we consider the sum of two cutting planes,which are designed respectively to the convx function and the augment function.By choosing appropriate descent condition,our method can keep track of the relationship between primal composite problem and approximate models.Under mild conditions,the convergence is proved and the accumulation point of iterations is the stationary point of the primal composite function.Two polynomial problems and twelve DC(difference of convex)problems are tested in numerical experiments.The preliminary numerical results show that the method proposed is effective and attractive.2.In Chapter 3,an adaptive proximal composite bundle method is proposed for a class of nonconvex and nonsmooth composite problem with inexact information.The composite problem is the sum of a finite convex function with inexact information and a nonconvex function.For the nonconvex function,we design the convexification technique and ensure the linearization errors of its augment function nonnegative.Then the sum of the convex function and the augment function is regarded as an approximate function to the primal problem.For the approximate function,we adopt disaggregate strategy and regard the sum of the cutting plane models of the convex function and the augment function as a a cutting plane model for the approximate function.Then we give the adaptive nonconvex proximal bundle method.Meanwhile,for the convex function with inexact information,we utilize the noise management strategy and update the proximal parameter to reduce the influence of inexact information.The method can obtain an approximate solution.The preliminary numerical results show our algorithm is effective and reliable.3.The bundle modification strategy was proposed for the convex unconstrained problems,its most interesting feature was the reduction of the calls for the quadratic programming solver.In Chapter 4,we extend the bundle modification strategy to a class of nonconvex nonsmooth constraint problems.Concretely,we adopt the convexification technique to the objective function and constraint function,take the penalty strategy to transfer the modified model into an unconstrained optimization and focus on the unconstrained problem with proximal bundle method and the bundle modification strategies.The global convergence of the corresponding algorithm is proved.The primal numerical results show that the proposed algorithms indeed reduce the calls for the quadratic programming solver and are promising and effective.4.Chapter 5 proposes a proximal bundle method based on the filter strategy for the nonsmooth constrained optimization.For the constrained problem,its objective function is lower-c2 and the constrained functions are convex.We adopt the convexification technique for the objective function and ensure the corresponding linearization error to be nonnegative.Then we design the exact penalty function to transfer the problem into an unconstrained model.For the model,a proximal bundle method with the filter strategy is proposed and numerical results indicate the algorithm is promising and reliable.