A DUALITY APPROACH AND OPTIMALITY CONDITIONS FOR SIMPLE CONVEX BILEVEL PROGRAMMING PROBLEMS

The paper deals with a convex bilevel programming problem (S) which never satisfies the Slater's condition. Using epsilon-approximate solutions of the lower level problem, we consider a regularized bilevel problem (S epsilon()) of (S) investigated by Lignola and Morgan (1997) that satisfies this condition. As approximation results, they obtained that inf S epsilon -> inf S when e goes to zero and that any accumulation point of a sequence of regularized solutions solves the original problem (S). Via the Fenchel-Lagrange duality, we provide optimality conditions for the regularized problem. Then, necessary optimality conditions are established for a class of solutions of problem (S). Finally, sufficient optimality conditions are established for (S).