Bilevel programming: reformulations, regularity, and stationarity

We have considered the bilevel programming problem in the case where the lower-level problem admits more than one optimal solution. It is well-known in the literature that in such a situation, the problem is ill-posed from the view point of scalar objective optimization. Thus the optimistic and pessimistic approaches have been suggested earlier in the literature to deal with it in this case. In the thesis, we have developed a unified approach to derive necessary optimality conditions for both the optimistic and pessimistic bilevel programs, which is based on advanced tools from variational analysis. We have obtained various constraint qualifications and stationarity conditions depending on some constructive representations of the solution set-valued mapping of the follower’s problem. In the auxiliary developments, we have provided rules for the generalized differentiation and robust Lipschitzian properties for the lower-level solution setvalued map, which are of a fundamental interest for other areas of nonlinear and nonsmooth optimization. Some of the results of the aforementioned theory have then been applied to derive stationarity conditions for some well-known transportation problems having the bilevel structure