The Research Into Optimality Sufficient Conditions Of Nondifferentiable Composite Multiobjective Programming

Multiobjective optimization is one important aspect in research of optimal theory and method.This paper is mainly to discuss many multiobjective optimization problems,Several optimality sufficient conditions of composite multiobjective programming have been established in this article.This paper has been divided into two parts for discussion, in part one , nonsmooth composite multiobjective programming with equality constraints and inequality constraints has been discussed in Banach space. In this part ,we have used Jacobian matrix to take the place of generalized gradient problem in nonsmooth optimization to discuss. At the same time,new conception of convexity has been given at conditions of nonsmooth composite optimization.Then we can get some optimality sufficient conditions when the objective function is quasi-convex.In part two, nonsmooth composite multiobjective programming has been discussed in Hilbert space, in this section, new projection operator is produced for original projection operator improving.And in this part, projection subset is divided into two parts which is convex set and polyhedron convex set.In this paper,we use projection gradient to make convergence sequence of points and give algorithm of convergence sequence of points.Then we prove that the sequence generated by this method is limited and convergent. At the same time,we also prove that the sequence generated is convergent to a minimizer.Finaly,we can get optimality conditions and its testify of nonsmooth composite multiobjective programming in Hilbert space.This paper greatly solves difficulties which is hard to calculate in computer for using generalized gradient to solve that problem because of the new research methods and technology. Meanwhile, it make theory and property of multiobjective optimization extend,which lay the theoretical foundation for this problem,and it has wide application in practical problem.