A block coordinate variable metric forward–backward algorithm

• Published in: Journal of global optimization Vol. 66; no. 3; pp. 457 - 485
• Main Authors: Chouzenoux, Emilie, Pesquet, Jean-Christophe, Repetti, Audrey
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2016; Springer; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Computer Science; Convergence; Convex analysis; Efficiency; Engineering Sciences; Inequalities; Mathematical analysis; Mathematics; Mathematics and Statistics; Minimization; Operations Research/Decision Theory; Optimization; Real Functions; Signal and Image processing; Strategy; Studies; 68U10; Nonconvex optimization; Phase retrieval; Proximity operator; Block coordinate descent; Alternating minimization; 90C26; 90C05; Inverse problems; 90C25; 94A08; 65K10; Nonsmooth optimization; 49M27; 65F08; Majorize–Minimize algorithm; Majorize-Minimize algorithm
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-016-0405-9

A number of recent works have emphasized the prominent role played by the Kurdyka-Łojasiewicz inequality for proving the convergence of iterative algorithms solving possibly nonsmooth/nonconvex optimization problems. In this work, we consider the minimization of an objective function satisfying this property, which is a sum of two terms: (i) a differentiable, but not necessarily convex, function and (ii) a function that is not necessarily convex, nor necessarily differentiable. The latter function is expressed as a separable sum of functions of blocks of variables. Such an optimization problem can be addressed with the Forward–Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize–Minimize principle. We propose to combine the latter acceleration technique with an alternating minimization strategy which relies upon a flexible update rule. We give conditions under which the sequence generated by the resulting Block Coordinate Variable Metric Forward–Backward algorithm converges to a critical point of the objective function. An application example to a nonconvex phase retrieval problem encountered in signal/image processing shows the efficiency of the proposed optimization method.