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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.