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.
Admissible control Averaged operator Fixed point iteration Local convergence Proximal algorithms Set-valued map
Mathematics Subject Classification
90C26 7H10 47H04
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The authors are thankful to the anonymous referee for their constructive comments and suggestions. MND was partially supported by the Australian Research Council (ARC) Discovery Project DP160101537 and by the Priority Research Centre for Computer-Assisted Research Mathematics and its Applications (CARMA) at the University of Newcastle. He wishes to acknowledge the hospitality and the support of D. Russell Luke during his visit to the Universität Göttingen. MKT was partially supported by a Postdoctoral Fellowship from the Alexander von Humboldt Foundation.
Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, Heidelberg (2012)zbMATHGoogle Scholar
Borwein, J.M., Tam, M.K.: The cyclic Douglas–Rachford method for inconsistent feasibility problems. J. Nonlinear Convex Anal. 16(4), 537–587 (2015)MathSciNetzbMATHGoogle Scholar
Bauschke, H.H., Noll, D., Phan, H.: Linear and strong convergence of algorithms involving averaged nonexpansive operators. J. Math. Anal. Appl. 421(1), 1–20 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
Bauschke, H.H., Dao, M.N.: On the finite convergence of the Douglas–Rachford algorithm for solving (not necessarily convex) feasibility problems in Euclidean spaces. SIAM J. Optim. 27(1), 507–537 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
Hesse, R., Luke, D.R., Neumann, P.: Alternating projections and Douglas–Rachford for sparse affine feasibility. IEEE Trans. Signal Process. 62(18), 4868–4881 (2014)MathSciNetCrossRefzbMATHGoogle Scholar