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The Alternating Minimization Algorithm (AMA) has been proposed by Tseng tosolve convex programming problems with two-block separable linear constraintsand objectives, whereby (at least) one of the components of the latter isassumed to be strongly convex. The fact that one of the subproblems to besolved within the iteration process of AMA does not usually correspond to thecalculation of a proximal operator through a closed formula, affects theimplementability of the algorithm. In this paper we allow in each block of theobjective a further smooth convex function and propose a proximal version ofAMA, called Proximal AMA, which is achieved by equipping the algorithm withproximal terms induced by variable metrics. For suitable choices of the latter,the solving of the two subproblems in the iterative scheme can be reduced tothe computation of proximal operators. We investigate the convergence of theproposed algorithm in a real Hilbert space setting and illustrate its numericalperformances on two applications in image processing and machine learning.

The Proximal Alternating Minimization Algorithm for two-block separable convex optimization problems with linear constraints