The Nonlinear Augmented Lagrangian Method Of Set-valued Optimization

In this paper, we introduce a new class of non-linear augmented Lagrangian penali-ty function with weak zero extreme property, and study a set-valued vector optimizationproblem in infinite-dimensional Banach spaces which have partial order by using theaugmented Lagrangian method, the concept of conjugate, bi-conjugate, abstract sub-gradient, the stability of primal problem and so on. One has that if the primal problemis stable, then exist strong duality between the primal problem and dual problem, weestablish some sufcient conditions, necessary and sufcient conditions for zero dualitygap between the primal problem and dual problem in the conditions of inf-externallystable under the framework of new class of non-linear augmented Lagrangian penal-ity function. Finally, we discuss the exact penalty representation and obtained thenecessary and sufcient conditions of exact penalty representation. These conclusionsextend some results for the real value optimization and set-value vctor optimizationproblem in the finite-dimsional spaces to Banach spaces.