Improvement Of Set Segmentation Method And Its Application In Nonconvex Optimization Problems

There are a large number of nonconvex optimization problems in the fields of economy,management and engineering technology.Therefore,solving nonconvex optimization problems has become a hot issue in the optimization field.In recent years,with a large number of in-depth research by scholars,many solutions have been proposed.Among them,the combined homotopy interior point method is one of the algorithms with global convergence.When the feasible region meets certain conditions,it can be solved by this method.For the smooth optimization problem on a non convex domain with holes,a set segmentation method is proposed,and the segmented sub problem can still be solved by the combined homotopy method.This paper studies a class of nonconvex and nonsmooth optimization problems,and gives a set segmentation method for the nonconvex region surrounded by piecewise smooth constraint function.Firstly,the nonconvex and nonsmooth original problem is divided into relatively independent subproblems,and the relationship between the subproblem under certain conditions and the KKT solution of the original problem is proved,so that the solution of the original problem can be obtained by finding the solution of the subproblem;Secondly,for the non convex and non smooth region obtained after segmentation,the aggregate method is used to smooth it,and the convergence of the aggregate homotopy method is proved.Finally,numerical examples are given,and the results show the effectiveness and feasibility of the method.Through the set segmentation method,the combined homotopy method can be applied to more complex nonconvex optimization problems,which expands its application range.