Semivectorial bilevel programming versus scalar bilevel programming. (English) Zbl 1453.49003
The core of this paper is the consideration of the relationship between a semivectorial bilevel programming problem and an associated scalar bilevel programming problem which is constructed by applying the weighted-sum-scalarization technique to the multi-objective lower level problem of the original problem. After giving the necessary notation in Section 2,
the problem and its surrogate are considered in Section 3. Then Section 4 discusses the precise relationship between semivectorial bilevel programming and scalar bilevel programming. This corrects the results from [S. Dempe et al., J. Optim. Theory Appl. 157, No. 1, 54–74 (2013; Zbl 1266.90160)] and provides some more insights to semivectorial bilevel programming. Finally, the existence of solutions in semivectorial bilevel programming is discussed in Section 5. After giving some theoretical investigations, two classes of examples from bilevel optimal control are discussed, namely simple semivectorial bilevel optimal control problems and inverse multi-objective optimal control problems. The results presented in this paper depict that surrogates of bilevel programming problems which are constructed by interpreting implicit multipliers as explicit variables have to be investigated with great care when local minimizers are considered.
Reviewer: Frank Werner (Magdeburg)
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49J27 | Existence theories for problems in abstract spaces |
| 90C29 | Multi-objective and goal programming |
| 90C48 | Programming in abstract spaces |
Citations:
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