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Parametrically prox-regular functions

Journal Article


Abstract


  • Prox-regularity is a generalization of convexity that includes all C2, lower-C2, strongly amenable, and primal-lower-nice functions. The study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-proxregular) 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 stability of minimizers in optimization problems. This document 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 nonconvex proximal average. We develop an alternate representation of a para-prox-regular function, related to the monotonicity of an f-attentive-localization as was done for prox-regular functions. This extends a result of Levy, who used an alternate approach to show one implication of the 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 is given, and a relaxation of its necessary conditions is presented.

Publication Date


  • 2014

Citation


  • Hare, W. & Planiden, C. (2014). Parametrically prox-regular functions. Journal of Convex Analysis, 21 (4), 901-923.

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1140

Number Of Pages


  • 22

Start Page


  • 901

End Page


  • 923

Volume


  • 21

Issue


  • 4

Place Of Publication


  • Germany

Abstract


  • Prox-regularity is a generalization of convexity that includes all C2, lower-C2, strongly amenable, and primal-lower-nice functions. The study of prox-regular functions provides insight on a broad spectrum of important functions. Parametrically prox-regular (para-proxregular) 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 stability of minimizers in optimization problems. This document 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 nonconvex proximal average. We develop an alternate representation of a para-prox-regular function, related to the monotonicity of an f-attentive-localization as was done for prox-regular functions. This extends a result of Levy, who used an alternate approach to show one implication of the 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 is given, and a relaxation of its necessary conditions is presented.

Publication Date


  • 2014

Citation


  • Hare, W. & Planiden, C. (2014). Parametrically prox-regular functions. Journal of Convex Analysis, 21 (4), 901-923.

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1140

Number Of Pages


  • 22

Start Page


  • 901

End Page


  • 923

Volume


  • 21

Issue


  • 4

Place Of Publication


  • Germany