Epi-convergence: The Moreau Envelope and Generalized Linear-Quadratic Functions

Chayne Planiden () and Xianfu Wang ()
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Chayne Planiden: University of Wollongong
Xianfu Wang: University of British Columbia Okanagan

Journal of Optimization Theory and Applications, 2018, vol. 177, issue 1, No 2, 63 pages

Abstract: Abstract This work explores the class of generalized linear-quadratic functions, constructed using maximally monotone symmetric linear relations. Calculus rules and properties of the Moreau envelope for this class of functions are developed. In finite dimensions, on a metric space defined by Moreau envelopes, we consider the epigraphical limit of a sequence of quadratic functions and categorize the results. We examine the question of when a quadratic function is a Moreau envelope of a generalized linear-quadratic function; characterizations involving nonexpansiveness and Lipschitz continuity are established. This work generalizes some results by Hiriart-Urruty and by Rockafellar and Wets.

Keywords: Attouch–Wets metric; Complete metric space; Epi-convergence; Extended seminorm; Fenchel conjugate; Firmly nonexpansive; Generalized linear-quadratic function; Linear relation; Lipschitz continuous; Maximally monotone; Nonexpansive; Moreau envelope; Proximal mapping; 47A06; 52A41; 47H05; 90C31 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-018-1254-0

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