Abstract: In this paper, we investigate a class of pessimistic bilevel programming problem, where the upper-level problem consists of a finite number of inequalities constraints and the lower-level problem is a cone constrained optimization problem. Firstly, using the maximization optimal value function of the upper-level problem and the minimization optimal value function of the lower-level problem, we translate the original problem into a single-level constrained optimization problem. Under some suitable assumptions, considering the subdifferential estimate of the maximization optimal value function and the properties of the upper and lower bound of the minimization optimal value functions directional derivative, we obtain the detailed first-order necessary optimality conditions for the original problem.

Key words: pessimistic bilevel programming problem, optimal value function method, constraint nondegeneracy condition, directional derivatives, Robinson constraint qualification, first-order necessary optimality conditions