A superlinearly convergent R-regularized Newton scheme for variational models with concave sparsity-promoting priors

Authors:
Michael Hintermüller;Tao Wu
Affiliations:
Department of Mathematics, Humboldt-University of Berlin, Berlin, Germany 10099;Institute for Mathematics and Scientific Computing, Karl-Franzens-University of Graz, Graz, Austria 8010
Venue:
Computational Optimization and Applications
Year:
2014

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Abstract

A general class of variational models with concave priors is considered for obtaining certain sparse solutions, for which nonsmoothness and non-Lipschitz continuity of the objective functions pose significant challenges from an analytical as well as numerical point of view. For computing a stationary point of the underlying variational problem, a Newton-type scheme with provable convergence properties is proposed. The possible non-positive definiteness of the generalized Hessian is handled by a tailored regularization technique, which is motivated by reweighting as well as the classical trust-region method. Our numerical experiments demonstrate selected applications in image processing, support vector machines, and optimal control of partial differential equations.