Generic property of the partial calmness condition for bilevel programming problems

TL;DR

This paper investigates the partial calmness condition in bilevel programming, establishing its generic nature under certain conditions and providing optimality criteria without extra qualifications, with implications for economics.

Contribution

It introduces a sufficient partial error bound condition guaranteeing partial calmness and proves its genericity for important economic applications.

Findings

Partial error bound condition is generic for combined programs.

Partial calmness is not a stringent assumption in key settings.

Derived optimality conditions match existing implicit forms.

Abstract

The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stationary conditions from a generic point of view. Our main result states that the partial error bound condition for the combined programs based on B and FJ conditions are generic for an important setting with applications in economics and hence the partial calmness for the combined program is not a particularly stringent assumption. Moreover we derive optimality conditions for the combined program for the generic case without any extra constraint qualifications and show the exact equivalence between our optimality condition and the one by Jongen and Shikhman given in implicit form. Our arguments are based on Jongen, Jonker and Twilt's generic (five type) classification of the so-called generalized critical points for one-dimensional parametric optimization problems and Jongen and Shikhman's generic local reductions of BLPPs.