Inexact alternating-direction-based contraction methods for separable linearly constrained convex optimization
Let \(b\in \mathbb{R}^{m}\) and, for \(i=1,\dots,N\), let \(\theta _{i}:\mathbb{R}^{n_{i}}\rightarrow \mathbb{R}\) be a convex function, \(A_{i}\in \mathbb{R}^{m\times n_{i}}\), and \(\mathcal{X}_{i}\subseteq \mathbb{R}^{n_{i}}\) be a closed convex set. The problem under consideration consists in minimizing \(\sum\limits_{i=1}^{N}\theta _{i}(x_{i})\) subject to \(\sum\limits_{i=1}^{N}A_{i}x_{i}=b\) and \(x_{i}\in \mathcal{X}_{i}\) \((i=1,\dots,N)\). The authors present two modifications of the method proposed by the second author et al. [Optimization 62, No. 4, 573--596 (2013; Zbl 1273.90122)], which have the advantage that the subproblems are solved inexactly. Convergence results are obtained, and the rates of convergence are proved to be \(O(1/t)\).
scientific article; zbMATH DE number 6363603
scientific article