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Enhanced Optimality Conditions and New Constraint Qualifications for Nonsmooth Optimization Problems

The main purpose of this dissertation is to investigate necessary optimality conditions for a class of very general nonsmooth optimization problems called the mathematical program with geometric constraints
(MPGC). The geometric constraint means that the image of certain mapping is included in a
nonempty and closed set.

We first study the conventional nonlinear program with equality, inequality and abstract set constraints as a special case of MPGC. We derive the enhanced Fritz John condition and from which, we obtain the enhanced Karush-Kuhn-Tucker (KKT) condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity implies the existence of a local error bound. We also give a tighter upper estimate for the Fr\'chet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers.

We then consider a more general MPGC where the image of the mapping from a Banach space is included in a
nonempty and closed subset of a finite dimensional space. We obtain the enhanced Fritz John necessary optimality conditions in terms of the
approximate subdifferential. One of the technical
difficulties in obtaining such a result in an infinite dimensional space is
that no compactness result can be used to show the existence of local
minimizers of a perturbed problem. We employ the celebrated
Ekeland's variational principle to obtain the results instead. We then apply our results to the study of exact penalty and sensitivity analysis.

We also study a special class of MPCG named mathematical programs with equilibrium constraints (MPECs). We argue that the MPEC-linear independence constraint qualification is not a constraint qualification for the strong (S-) stationary condition when the objective function is nonsmooth. We derive the enhanced Fritz John Mordukhovich (M-) stationary condition for MPECs. From this enhanced Fritz John M-stationary condition we introduce the associated MPEC generalized pseudonormality and quasinormality condition and build the relations between them and some other widely used MPEC constraint qualifications. We give upper estimates for the subdifferential of the value function in terms of the enhanced M- and C-multipliers respectively.

Besides, we focus on some new
constraint qualifications introduced for nonlinear extremum problems in the
recent literature. We show that, if the constraint functions are continuously
differentiable, the relaxed Mangasarian-Fromovitz constraint qualification (or,
equivalently, the constant rank of the subspace component condition) implies
the existence of local error bounds. We further extend the new result to the MPECs. / Graduate / 0405

Identiferoai:union.ndltd.org:uvic.ca/oai:dspace.library.uvic.ca:1828/5762
Date12 December 2014
CreatorsZhang, Jin
ContributorsYe, Juan Juan
Source SetsUniversity of Victoria
LanguageEnglish, English
Detected LanguageEnglish
TypeThesis
RightsAvailable to the World Wide Web, http://creativecommons.org/licenses/by-nc-nd/2.5/ca/

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