Directional Necessary Optimality Conditions for Bilevel Programs

Kuang Bai () and Jane J. Ye ()
Additional contact information
Kuang Bai: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Jane J. Ye: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada

Mathematics of Operations Research, 2022, vol. 47, issue 2, 1169-1191

Abstract: The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian–Fromovitz constraint qualification (MFCQ). In this paper, we show that even the first order sufficient condition for metric subregularity (which is, in general, weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of a directional calmness condition and show that, under the directional calmness condition, the directional necessary optimality condition holds. Although the directional optimality condition is, in general, sharper than the nondirectional one, the directional calmness condition is, in general, weaker than the classical calmness condition and, hence, is more likely to hold. We perform the directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.

Keywords: Primary: 90C46; secondary: 90C26; 90C30; 90C31; bilevel programs; constraint qualifications; necessary optimality conditions; directional derivatives; directional subdifferentials; directional quasi-normality (search for similar items in EconPapers)
Date: 2022
References: Add references at CitEc
Citations:

Downloads: (external link)
http://dx.doi.org/10.1287/moor.2021.1164 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:2:p:1169-1191

Access Statistics for this article

More articles in Mathematics of Operations Research from INFORMS Contact information at EDIRC.
Bibliographic data for series maintained by Chris Asher ().