Interior-Exterior Mixed Penalty Approach of Constrained Stackelberg Problems with Nonunique Lower Level Solutions

IN this paper, we consider a two-level optimization problem (S) (weak Stackelberg problem) in which the constraints of the upper level problem depend on the set of optimal solutions of the lower level problem, supposed not necessarily a singleton. Using penalty methods, we give an approximation of (S) by a sequence of one-level unconstrained optimization problems. Then, under appropriate assumptions, we prove that any sequence of optimal solutions to the penalized problems has accumulation points and any one of which is a solution to the problem (S).