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On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces

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In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective… Click to show full abstract

In this paper, the convergence of the fundamental alternating minimization is established for non-smooth non-strongly convex optimization problems in Banach spaces, and novel rates of convergence are provided. As objective function a composition of a smooth, and a block-separable, non-smooth part is considered, covering a large range of applications. For the former, three different relaxations of strong convexity are considered: (i) quasi-strong convexity; (ii) quadratic functional growth; and (iii) plain convexity. With new and improved rates benefiting from both separate steps of the scheme, linear convergence is proved forĀ (i) andĀ (ii), whereas sublinear convergence is showed for (iii).

Keywords: non smooth; convergence; non strongly; optimization; alternating minimization; smooth non

Journal Title: Optimization Letters
Year Published: 2022

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