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Parametrically prox-regular functions Planiden, Chayne
Abstract
Prox-regularity is a generalization of convexity that includes all lower-C² functions. Therefore, the study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-prox-regular) functions are a further extension of this family, produced by adding a parameter. Such functions have been shown to play a key role in understanding the stability of minimizers in optimization problems. This thesis discusses para-prox-regular functions in ℝn. We begin with some basic examples of para-prox-regular functions, and move on to the more complex examples of the convex and non-convex proximal averages. We develop an alternate representation of para-prox-regular functions, related to the monotonicity of an f-attentive ε-localization as was done for prox-regular functions [25]. Levy in [18] provided proof of one implication of this relationship; we provide a characterization. We analyze two common forms of parametrized functions that appear in optimization: finite parametrized sum of functions, and finite parametrized max of functions. The example of strongly amenable functions by Poliquin and Rockafellar [27] is given, and a relaxation of its necessary conditions is presented. Some open questions and directions of further research are stated.
Item Metadata
Title |
Parametrically prox-regular functions
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2013
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Description |
Prox-regularity is a generalization of convexity that includes all lower-C² functions. Therefore, the study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-prox-regular) functions are a further extension of this family, produced by adding a parameter. Such functions have been shown to play a key role in understanding the stability of minimizers in optimization problems. This thesis discusses para-prox-regular functions in ℝn.
We begin with some basic examples of para-prox-regular functions, and move on to the more complex examples of the convex and non-convex proximal averages. We develop an alternate representation of para-prox-regular functions, related to the monotonicity of an f-attentive ε-localization as was done for prox-regular functions [25]. Levy in [18] provided proof of one implication of this relationship; we provide a characterization. We analyze two common forms of parametrized functions that appear in optimization: finite parametrized sum of functions, and finite parametrized max of functions. The example of strongly amenable functions by Poliquin and Rockafellar [27] is given, and a relaxation of its necessary conditions is presented. Some open questions and directions of further research are stated.
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Genre | |
Type | |
Language |
eng
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Date Available |
2013-04-12
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0073783
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URI | |
Degree | |
Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2013-05
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Campus | |
Scholarly Level |
Graduate
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Rights URI | |
Aggregated Source Repository |
DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International