Proximal Alternating Direction Algorithm For A Kind Of Nonsmooth Optimization Problem

In real life,many application problem can be abstractly described by non-smooth convex functions,such as image compression,signal processing,matrix decomposition,sparse signal recovery,etc.These problems can be regarded as minimizing the sum of several of convex or nonconvex functions in real value space.As a result,it has theoretical and practical value to study the non-smooth optimization problem with Structural function.The problem model contains both the objection and constraints.If there is a non-convex function,the problem is non-convex optimization,otherwise it is convex optimization.In this paper,we study a class of unconstrained non-smooth optimization problems with structuralfeatures.Thestructureoftheobjectivefunctionis??x,y?=f?x?+g?y?+h?x,y?,Attouch^[1]and Bolte^[2]et al.studied such structural problems.According to the convexity of the problem,this paper starts from two aspects:one is to solve the convex problem,the other is to solve the non-convex optimization problem.In this paper,two kinds of nonsmooth optimization are solved by using proximal alternating method.For nonsmooth and convex optimization problem,the objective function is described as sum of there convex function,where f and g are continuous convex functions,h is a continuous differentiable convex function.The partial derivatives of h are respectively satisfied Lipschitz conditions for x and y.x and y are approximated by quadratic regularization.The original problem is transformed into two convex subproblems by the classical Guass-Seidel iterative method.Then the two subproblems are alternately minimized.Hence,a new algorithm is proposed for this kind of structural optimization problem,which is called the secondary upper-bound inexact proximal alternating algorithm?SUIPAD?.For nonconvex and nonsmooth optimization problem,the objective function is described as the sum of these convex and nonconvex functions,where f and g are continuous convex functions,h is a second differentiable function with Lipschitz gradient.Solving this kind of nonsmooth and nonconvex problem is to transform the original problem into two subproblems which are approximated by the iterative method of Guass-Seidel iteration.h is linearly dependent on the subgradient of x and y,and the second proximal terms of variables x and y are added to the subproblem respectively.In this paper,a new proximal alternate method is proposed for the nonsmooth and nonconvex problem under reasonable conditions by proximal operator and alternate direction method,namely the quadratic upper bound approximation algorithm?QUA?.Our main results are as follows:In both convex and non-convex cases,a kind of structural optimization problems are solved by the presented algorithms.For convex objective function,every limit point of sequence generated by SUIPAD iterative algorithm is the global optimal solution of the original problem.For nonconvex objective function,every limit point of arbitrary bounded sequence generated by QUA iterative algorithm is the critical point of the original problem.