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Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization
We study the extension of the Chambolle--Pock primal-dual algorithm to
nonsmooth optimization problems involving nonlinear operators between function
spaces. Local convergence is shown under technical conditions including metric
regularity of the corresponding primal-dual optimality conditions. We also show
convergence for a Nesterov-type accelerated variant provided one part of the
functional is strongly convex.
We show the applicability of the accelerated algorithm to examples of inverse
problems with - and -fitting terms as well as of
state-constrained optimal control problems, where convergence can be guaranteed
after introducing an (arbitrary small, still nonsmooth) Moreau--Yosida
regularization. This is verified in numerical examples
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