Abstract
The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of a proximal operator through a closed formula affects the implementability of the algorithm. In this paper, we allow in each block of the objective a further smooth convex function and propose a proximal version of the algorithm, which is achieved by equipping the algorithm with proximal terms induced by variable metrics. For suitable choices of the latter, the solving of the two subproblems in the iterative scheme can be reduced to the computation of proximal operators. We investigate the convergence of the proposed algorithm in a real Hilbert space setting and illustrate its numerical performances on two applications in image processing and machine learning.
Highlights
Tseng introduced in [1] the so-called Alternating Minimization Algorithm (AMA) to solve optimization problems with two-block separable linear constraints and two nonsmooth convex objective functions, one of these assumed to be strongly convex
We address in a real Hilbert space setting a more involved two-block separable optimization problem, which is obtained by adding in each block of the objective a further smooth convex function
We propose a so-called Proximal Alternating Minimization Algorithm (Proximal AMA), which is obtained by inducing in each of the minimization subproblems additional proximal terms defined by means of positively semidefinite operators
Summary
Tseng introduced in [1] the so-called Alternating Minimization Algorithm (AMA) to solve optimization problems with two-block separable linear constraints and two nonsmooth convex objective functions, one of these assumed to be strongly convex. The strong convexity of one of the objective functions allows to reduce the corresponding minimization subproblem to the calculation of the proximal operator of a proper, convex and lower semicontinuous function This is for the second minimization problem in general not the case; with the exception of some very particular cases, one has to use a subroutine in order to compute the corresponding iterate. We address in a real Hilbert space setting a more involved two-block separable optimization problem, which is obtained by adding in each block of the objective a further smooth convex function To solve this problem, we propose a so-called Proximal Alternating Minimization Algorithm (Proximal AMA), which is obtained by inducing in each of the minimization subproblems additional proximal terms defined by means of positively semidefinite operators. Parts of the convergence analysis for the Proximal AMA are carried out in a similar spirit to the convergence proofs in these papers
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