Optimality conditions for the simple convex bilevel programming problem in Banach spaces

Abstract

The simple convex bilevel programming problem is a convex minimization problem whose feasible set is the solution set of another convex optimization problem. Such problems appear frequently when searching for the projection of a certain point onto the solution set of another program. Due to the nature of the problem, Slater’s constraint qualification generally fails to hold at any feasible point. Hence, one has to formulate weaker constraint qualifications or stationarity notions in order to state optimality conditions. In this paper, we use two different single-level reformulations of the problem, the optimal value and the Karush–Kuhn–Tucker approach, to derive optimality conditions for the original program. Since all these considerations are carried out in Banach spaces, the results are not limited to standard optimization problems in . On the road, we introduce and discuss a certain concept of M-stationarity for mathematical programs with complementarity constraints in Banach spaces.

Keywords:

• Bilevel programming
• constraint qualifications
• convex programming
• mathematical program with complementarity constraints
• programming in Banach spaces

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Erratum

Acknowledgements

The authors would like to thank Gerd Wachsmuth for some fruitful discussion which led to the result postulated in Lemma 4.3. Furthermore, we would like to thank three anonymous reviewers for their valuable comments which helped us to improve the quality of the manuscript.

Notes

No potential conflict of interest was reported by the authors.

Dedicated to Professor Stephan Dempe on the occasion of his 60th birthday.

This article was originally published with errors. This version has been corrected. Please see Erratum (https://doi.org/10.1080/02331934.2017.1405156)

Additional information

Funding

The research of the first author is supported by the Deutsche Forschungsgemeinschaft within the [project DE 650/7-1] Lösungsalgorithmen für Zwei-Ebenen-Optimierungsaufgaben.