Optimization With Cone Constraints In Banach Space

In this paper, we will consider the optimization problem with coneconstrains in Banach space. By introducing a class of nonlinear aug-mented lagrangian functions satisfying weak peak at zero property andthe-perturbation conditions in Banach space, we proof these conditionsare necessary and sufcient conditions for the zero duality gap propertybetween the primal problem and the corresponding nonlinear augmentedlagrangian dual problem. We firstly introduced sub-coercivity conditionto nonlinear augmented penalty function. At the same time, we removethe condition which nonlinear augmented penalty function is monoton-ically nondecreasing on the penalty factor outside some neighborhoodof an origin. Under weaker conditions, we also obtain the zero dual-ity gap property. Finally, we show the existence of sub-optimal pathof lagrangian optimization problem (L). and the set of cluter points ofthe sub-optimal path contained in the optimal solution set of the primalproblem (P).