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Convex Analysis in Hadamard Spaces

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VMVW01 - Variational methods, new optimisation techniques and new fast numerical algorithms

joint work with M. Bacak, R. Bergmann, M. Montag and J. Persch

The aim of the talk is two-fold:

1. A well known result of H. Attouch states that the Mosco convergence of a sequence of proper convex lower semicontinuous functions defined on a Hilbert space is equivalent to the pointwise convergence of the associated Moreau envelopes. In the present paper we generalize this result to Hadamard spaces. More precisely, while it has already been known that the Mosco convergence of a sequence of convex lower semicontinuous functions on a Hadamard space implies the pointwise convergence of the corresponding Moreau envelopes, the converse implication was an open question. We now fill this gap.  Our result has several consequences. It implies, for instance, the equivalence of the Mosco and Frolik-Wijsman convergences of convex sets. As another application, we show that there exists acomplete metric on the cone of proper convex lower semicontinuous functions on a separable Hadamard space such that asequence of functions converges in this metric if and only if it converges in the sense of Mosco.

2. We extend the parallel Douglas-Rachford algorithm  to the manifold-valued setting.

This talk is part of the Isaac Newton Institute Seminar Series series.

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