Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms

Peter Ochs (), Jalal Fadili () and Thomas Brox ()
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Peter Ochs: Saarland University
Jalal Fadili: Normandie Université ENSICAEN, CNRS, GREYC
Thomas Brox: University of Freiburg

Journal of Optimization Theory and Applications, 2019, vol. 181, issue 1, No 13, 244-278

Abstract: Abstract We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error. Special instances of the algorithm with a Euclidean distance function are, for example, gradient descent, forward–backward splitting, ProxDescent, without the common requirement of a “Lipschitz continuous gradient”. In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions), replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and nonlinear inverse problems in signal/image processing and machine learning.

Keywords: Bregman minimization; Legendre function; Model function; Growth function; Non-convex non-smooth; Abstract algorithm; 49J52; 65K05; 65K10; 90C26 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10957-018-01452-0

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