Solving ill-posed bilevel programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem an a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem

Variational Analysis in Bilevel Programming

The paper is devoted to applications of advanced tools of modern variational analysis and generalized differentiation to problems of optimistic bilevel programming. In this way, new necessary optimality conditions are derived for two major classes of bilevel programs: those with partially convex and with fully convex lower-level problems. We provide detailed discussions of the results obtained and their relationships with known results in this area

Tropical analogues of a Dempe-Franke bilevel optimization problem

We consider the tropical analogues of a particular bilevel optimization problem studied by Dempe and Franke and suggest some methods of solving these new tropical bilevel optimization problems. In particular, it is found that the algorithm developed by Dempe and Franke can be formulated and its validity can be proved in a more general setting, which includes the tropical bilevel optimization problems in question. We also show how the feasible set can be decomposed into a finite number of tropical polyhedra, to which the tropical linear programming solvers can be applied.Comment: 11 pages, 1 figur

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page