Directional constraint qualifications and optimality conditions with application to bilevel programs

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dc.contributor.authorBai, Kuang
dc.contributor.supervisorYe, Juan Juan
dc.date.accessioned2020-07-18T23:52:13Z
dc.date.copyright2020en_US
dc.date.issued2020-07-18
dc.degree.departmentDepartment of Mathematics and Statistics
dc.degree.levelDoctor of Philosophy Ph.D.en_US
dc.description.abstractThe main purpose of this dissertation is to investigate directional constraint qualifications and necessary optimality conditions for nonsmooth set-constrained mathematical programs. First, we study sufficient conditions for metric subregularity of the set-constrained system. We introduce the directional version of the quasi-/pseudo-normality as a sufficient condition for metric subregularity, which is weaker than the classical quasi-/pseudo-normality, respectively. Then we apply our results to complementarity and Karush-Kuhn-Tucker systems. Secondly, we study directional optimality conditions of bilevel programs. It is well-known that the value function reformulation of bilevel programs provides equivalent single-level optimization problems， which are nonsmooth and never satisfy the usual constraint qualifications such as the Mangasarian-Fromovitz constraint qualification (MFCQ). We show that even the first-order sufficient condition for metric subregularity (which is generally weaker than MFCQ) fails at each feasible point of bilevel programs. We introduce the directional Clarke calmness condition and show that under the directional Clarke calmness condition, the directional necessary optimality condition holds. We perform directional sensitivity analysis of the value function and propose the directional quasi-normality as a sufficient condition for the directional Clarke calmness.en_US
dc.description.embargo2021-07-07
dc.description.scholarlevelGraduateen_US
dc.identifier.bibliographicCitationK. Bai, J. J. Ye and J. Zhang, Directional quasi-/pseudo-normality as sufficient conditions for metric subregularity, SIAM J. Optim., 29 (2019), pp. 2625-2649.en_US
dc.identifier.bibliographicCitationK. Bai and J. J. Ye, Directional necessary optimality conditions for bilevel programs, submitted to Math. Oper. Res., 2020. arXiv: 2004.01783.en_US
dc.identifier.urihttp://hdl.handle.net/1828/11939
dc.languageEnglisheng
dc.language.isoenen_US
dc.rightsAvailable to the World Wide Weben_US
dc.subjectdirectional limiting normal conesen_US
dc.subjectmetric subregularityen_US
dc.subjecterror boundsen_US
dc.subjectcalmnessen_US
dc.subjectdirectional quasi-normalityen_US
dc.subjectdirectional pseudo-normalityen_US
dc.subjectcomplementarity systemsen_US
dc.subjectbilevel programsen_US
dc.subjectnecessary optimality conditionsen_US
dc.subjectdirectional subdifferentialen_US
dc.subjectsensitivity analysisen_US
dc.titleDirectional constraint qualifications and optimality conditions with application to bilevel programsen_US
dc.typeThesisen_US

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