Prox-regular functions in variational analysis

Poliquin, R.; Rockafellar, R.

doi:10.1090/S0002-9947-96-01544-9

Prox-regular functions in variational analysis
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by R. A. Poliquin and R. T. Rockafellar

Trans. Amer. Math. Soc. 348 (1996), 1805-1838

DOI: https://doi.org/10.1090/S0002-9947-96-01544-9

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Abstract:

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-$\mathcal {C}^{2}$ functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

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