Union Averaged Operators with Applications to Proximal Algorithms for Min-Convex Functions

Minh N. Dao () and Matthew K. Tam ()
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Minh N. Dao: University of Newcastle
Matthew K. Tam: Universität Göttingen

Journal of Optimization Theory and Applications, 2019, vol. 181, issue 1, No 3, 94 pages

Abstract: Abstract In this paper, we introduce and study a class of structured set-valued operators, which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations, where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions, where the first is convex and the second can be expressed as the minimum of finitely many convex functions.

Keywords: Admissible control; Averaged operator; Fixed point iteration; Local convergence; Proximal algorithms; Set-valued map; 90C26; 7H10; 47H04 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10957-018-1443-x

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