Abstract

Prox-regularity is a generalization of convexity that includes all C2\mathcal{C}^2, lower-C2\mathcal{C}^2, 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-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 stability of minimizers in optimization problems. This document discusses para-prox-regular functions in Rn\mathbb{R}^n. 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 ε-localization as was done for prox-regular functions by R. A. Poliquin and R. T. Rockafellar ["Prox-regular functions in variational analysis", Trans. Amer. Math. Soc. 348(5) (1996) 1805–1838]. This extends a result of A. B. Levy ["Calm minima in parameterized finite-dimensional optimization", SIAM J. Optim. 11(1) (2000) 160–178 (electronic)], 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 R. A. Poliquin and R. T. Rockafellar ["A calculus of prox-regularity", J. Convex Analyis 17(1) (2010) 203–210] is given, and a relaxation of its necessary conditions is presented

Author information

Contact details are reproduced from the original publication and may be historical.

Warren L. Hare

University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada

warren.hare@ubc.ca

Chayne Planiden

University of British Columbia, Okanagan Campus, 3333 University Way, Kelowna, BC, Canada

chayneplaniden@hotmail.com

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Suggested citation

W. L. Hare, C. Planiden. “Parametrically Prox-Regular Functions.” Journal of Convex Analysis 21 (2014), No. 4, 901–923. https://doi.org/10.68381/jca21049

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.