Directional Necessary Optimality Conditions for Bilevel Programs

• 成果类型: Article
• 署名作者: Bai, Kuang; Ye, Jane J.
• 署名单位: Hong Kong Polytechnic University; University of Victoria
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.1164
• 发表日期: 2022
• 页码: 1169-1191
• 关键词: mathematical programs; optimization problems; constraint systems; marginal function; calculus; calmness

摘要

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 non-smooth 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 conditionmay hold for the bilevel program.