Abstract
We consider bilevel optimization from the optimistic point of view. Let the pair (x, y) denote the variables. The main difficulty in studying such problems lies in the fact that the lower level contains a global constraint. In fact, a point (x, y) is feasible if y solves a parametric optimization problem L(x). In this paper we restrict ourselves to the special case that the variable x is one-dimensional. We describe the generic structure of the feasible set M. Moreover, we discuss local reductions of the bilevel problem as well as corresponding optimality criteria. Finally, we point out typical problems that appear when trying to extend the ideas to higher dimensional x-dimensions. This will clarify the high intrinsic complexity of the general generic structure of the feasible set M and corresponding optimality conditions for the bilevel problem U.
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Jongen, H.T., Shikhman, V. Bilevel optimization: on the structure of the feasible set. Math. Program. 136, 65–89 (2012). https://doi.org/10.1007/s10107-012-0551-x
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DOI: https://doi.org/10.1007/s10107-012-0551-x
Keywords
- Bilevel programming
- Parametric optimization
- Structure of the feasible set
- Local reduction
- Optimality criteria