RISK-AVERSE MODELS IN BILEVEL STOCHASTIC LINEAR PROGRAMMING

We consider a two-stage stochastic bilevel linear program where the leader contemplates the follower's reaction at the second stage optimistically. In this setting, the leader's objective function value can be modeled by a random variable, which we evaluate based on some law-invariant (quasi-)convex risk measure. After establishing Lipschitzian properties and existence results, we derive sufficient conditions for differentiability when the choice function is a Lipschitzian transformation of the expectation. This allows us to formulate first-order necessary optimality conditions for models involving certainty equivalents or expected disutilities. Moreover, a qualitative stability result under perturbation of the underlying probability distribution is presented. Finally, for finite discrete distributions, we reformulate the bilevel stochastic problems as standard bilevel problems and propose a regularization scheme for solving a deterministic bilevel programming problem.