A block coordinate variable metric forward–backward algorithm

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...

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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
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Abstract	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.
AbstractList	A number of recent works have emphasized the prominent role played by the Kurdyka-Lojasiewicz 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. A number of recent works have emphasized the prominent role played by the Kurdyka-ojasiewicz 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. A number of recent works have emphasized the prominent role played by the Kurdyka-Lojasiewicz 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 a non necessarily convex differentiable function and a non necessarily differentiable or convex function. 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. A number of recent works have emphasized the prominent role played by the Kurdyka-Aaojasiewicz 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. 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.
Audience	Academic
Author	Chouzenoux, Emilie Pesquet, Jean-Christophe Repetti, Audrey
Author_xml	– sequence: 1 givenname: Emilie surname: Chouzenoux fullname: Chouzenoux, Emilie email: emilie.chouzenoux@univ-mlv.fr organization: Laboratoire d’Informatique Gaspard Monge and CNRS UMR 8049, Université Paris-Est Marne-la-Vallée – sequence: 2 givenname: Jean-Christophe surname: Pesquet fullname: Pesquet, Jean-Christophe organization: Laboratoire d’Informatique Gaspard Monge and CNRS UMR 8049, Université Paris-Est Marne-la-Vallée – sequence: 3 givenname: Audrey surname: Repetti fullname: Repetti, Audrey organization: Institute of Sensors, Signals and Systems, Heriot-Watt University
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Keywords	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
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Snippet	A number of recent works have emphasized the prominent role played by the Kurdyka-Łojasiewicz inequality for proving the convergence of iterative algorithms... A number of recent works have emphasized the prominent role played by the Kurdyka-Lojasiewicz inequality for proving the convergence of iterative algorithms... A number of recent works have emphasized the prominent role played by the Kurdyka-ojasiewicz inequality for proving the convergence of iterative algorithms... A number of recent works have emphasized the prominent role played by the Kurdyka-Aaojasiewicz inequality for proving the convergence of iterative algorithms...
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Title	A block coordinate variable metric forward–backward algorithm
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