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Title: Polar convolution
Authors: Friedlander, MP
MacEdo, I
Pong, TK 
Issue Date: 2019
Source: SIAM journal on optimization, 2019, v. 29, no. 2, p. 1366-1391
Abstract: The Moreau envelope is one of the key convexity-preserving functional operations in convex analysis, and it is central to the development and analysis of many approaches for convex optimization. This paper develops the theory for an analogous convolution operation, called the polar envelope, specialized to gauge functions. Many important properties of the Moreau envelope and the proximal map are mirrored by the polar envelope and its corresponding proximal map. These properties include smoothness of the envelope function, uniqueness, and continuity of the proximal map, which play important roles in duality and in the construction of algorithms for gauge optimization. A suite of tools with which to build algorithms for this family of optimization problems is thus established.
Keywords: Gauge optimization
Max convolution
Proximal algorithms
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/18M1209088
Rights: © 2018 Society for Industrial and Applied Mathematics.
Posted with permission of the publisher.
The following publication Friedlander, M. P., Macêdo, I., & Pong, T. K. (2019). Polar convolution. SIAM Journal on Optimization, 29(2), 1366-1391 is available at
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