A Telescopic Bregmanian Proximal Gradient Method Without the Global Lipschitz Continuity Assumption

Daniel Reem (), Simeon Reich () and Alvaro Pierro ()
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Daniel Reem: The Technion - Israel Institute of Technology
Simeon Reich: The Technion - Israel Institute of Technology
Alvaro Pierro: CNPq

Journal of Optimization Theory and Applications, 2019, vol. 182, issue 3, No 1, 884 pages

Abstract: Abstract The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward–backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. In this paper, we discuss, in a wide class of finite- and infinite-dimensional spaces, a new variant of the proximal gradient method, which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain telescopic decomposition of the constraint set into subsets. Moreover, we use a Bregman divergence in the proximal forward–backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest.

Keywords: Bregman divergence; Lipschitz continuity; Minimization; TEPROG; Telescopic proximal gradient method; Strongly convex; 90C25; 49M27; 47J25; 90C30; 54C30; 26B25 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-019-01509-8

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