1. The Proximal Alternating Minimization Algorithm for Two-Block Separable Convex Optimization Problems with Linear Constraints
- Author
-
Radu Ioan Boţ, Ernö Robert Csetnek, Sandy Bitterlich, and Gert Wanka
- Subjects
Mathematical optimization ,Control and Optimization ,Computation ,0211 other engineering and technologies ,65K05 ,010103 numerical & computational mathematics ,02 engineering and technology ,Subderivative ,Management Science and Operations Research ,01 natural sciences ,Article ,90C25 ,Separable space ,symbols.namesake ,Fenchel duality ,Operator (computer programming) ,FOS: Mathematics ,Proximal AMA ,Lagrangian ,Saddle points ,Subdifferential ,Convex optimization ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,021103 operations research ,47H05 ,Applied Mathematics ,Hilbert space ,Numerical Analysis (math.NA) ,47H05, 65K05, 90C25 ,Optimization and Control (math.OC) ,Theory of computation ,symbols ,Convex function - 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. peerReviewed
- Published
- 2018