#### Document Type

Technical Report

#### Abstract

This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization spread under the name of the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that, although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do not to a large extent differ from those known for the conventional problem. It has been already well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This occurs to be related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the two level value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data.

#### Number in Series

2011.11

#### Disciplines

Applied Mathematics | Mathematics

#### AMS Subject Classification

90C26, 90C30-31, 90C46, 91C12, 91A65

#### Recommended Citation

Dempe, S; Mordukhovich, Boris S.; and Zemkoho, B, "Sensitivity Analysis For Two-Level Value Functions With Applications to Bilevel Programming" (2011). *Mathematics Research Reports*. 92.

https://digitalcommons.wayne.edu/math_reports/92

## Comments

The authors acknowledge partial support by the USA National Science Foundation under grant DMS-1007132, by the European Regional Development Fund (FEDER), and by the following Portuguese agencies: Foundation for Science and Technologies (FCT), Operational Program for Competitiveness Factors (COMPETE), and Strategic Reference Framework (QREN); and by the Deutscher Akademischer Austausch Dienst (DAAD).