Some New Algorithms For Convex Composite Optimization Problems

Convex composite optimization problems have widely applied in many fields Many optimization problems,such as minimax optimization problem,constrained optimization problems where the objective function is optimization problem of convex functions can all be regarded as a special case of convex composite optimization problems.Many practical optimization models,such as location problem,transportation problem and economics problem,involve compound convex function.So the convex composite model provides a unified framework for many optimization problems.In this paper,we present a new algorithm to solve the convex composite optimization problems,it’s called Newton projection method.And,we prove the algorithm own good global convergence.The main idea of this algorithm is to find the optimal solution by combination of Newton method and projection technique.In each iteration,first of all,we use the Newton step,right after we use the line search direction when the Newton step cannot satisfy certain restrictive condition.Finally,the method of projection cut down the distance between the iteration point and the solution set of the problem.This provides a new way to solve the complex convex optimization problems.The problem of limited termination of the feasible solution sequence generated by the algorithm has been widely concerned and studied for a long time.The concept of strong non-degenerate solution sets also plays an important role in the finite termination of feasible solution sequences.Burke and Ferris gave sufficient and necessary conditions for a feasible solution point sequence to have a finite termination under the assumption that the solution set satisfies the strong and weak minimum for convex programming.However,we find that its conclusion is not valid for non-convex optimization problems,so this paper studies the finite termination condition of feasible solution sequences for non-convex optimization problems.