Abstract
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 and 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.
Highlights
Most solution methods available in the current literature on bilevel optimization are designed only for well-posed problems, i.e., for the case where the reaction of the lowerlevel player is restricted to a unique optimal solution for each strategy of the upper-level player; see, e.g., [5, 7, 8, 39]
MinF (x, y(x)) s.t. x ∈ X, x where F and X denote the upper-level objective function and feasible set, respectively, while y(x) stands for the optimal solution of the lower-level optimization problem, which can be formulated as minf (x, y) s.t. y ∈ K(x)
The aim of the paper was to provide a first step towards solving the original optimistic bilevel program (Po) using a set-valued optimization technique, which is an extension of the implicit function approach in (Pi)
Summary
Most solution methods available in the current literature on bilevel optimization are designed only for well-posed problems, i.e., for the case where the reaction of the lowerlevel player is restricted to a unique optimal solution for each strategy of the upper-level player; see, e.g., [5, 7, 8, 39]. We consider the alternative approach to deal with ill-posed bilevel programs which consists to insert the lower-level solution mapping (1.4) in the upper-level objective function. The result clearly indicates that solving problem (Po) will provide a new direction to develop solution methods for the pessimistic bilevel program (Pp), which is the most difficult class of ill-posed bilevel programs Further recall that it was already shown in [14] that the optimality conditions of (Pp) can be obtained from (Po). We derive completely new results for parametric set-valued optimization, semivectorial bilevel optimization and multiobjective bilevel programs with vector-valued upper-and lower-level objective functions. The examples considered contribute to a better understanding of the models discussed above
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