Abstract
A number of recent works have emphasized the prominent role played by the KurdykaŁojasiewicz 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.
Original language  English 

Pages (fromto)  129 
Number of pages  29 
Journal  Journal of Global Optimization 
Early online date  10 Feb 2016 
DOIs  
Publication status  Epub ahead of print  10 Feb 2016 
Keywords
 Alternating minimization
 Block coordinate descent
 Inverse problems
 Majorize–Minimize algorithm
 Nonconvex optimization
 Nonsmooth optimization
 Phase retrieval
 Proximity operator
ASJC Scopus subject areas
 Computer Science Applications
 Control and Optimization
 Applied Mathematics
 Management Science and Operations Research
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Audrey Repetti
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Associate Professor
Person: Academic (Research & Teaching)