VOCAL 2024
Abstract Submission

62. Convergence of semi-convex functions in CAT(1) spaces and refelction on Douglas Rachford Operator Splitting Algorithm

Invited abstract in session TB-3: Topics in nonlinear programming, stream contributed papers.

Thursday, 10:00 - 11:30
Room: C 104

Authors (first author is the speaker)

1. Hedvig Gal
Department of Economics, Corvinus University of Budapest
2. Miklós Pálfia
Department of Mathematics, Corvinus University of Budapest

Abstract

The research presents the theory of gradient flows of semi-convex functions established in CAT(1)-spaces, which refers to the a priori estimation of Ohta and Pálfia in CAT(1)-spaces. It denotes that the commutativity property and semi-convexity of the squared distance function is enough to establish the uniqueness, the Evolution Variational Inequalities (EVI) and the contractivity of the gradient flow, similarly as in the CAT(0) setting using the Moreau-Yoshida resolvent. It considers that the discrete and continuous time flows are leading to a Law of Large Numbers in CAT(1)-spaces. The Mosco convergence was previously established for CAT(0) spaces by Kuwae-Shioya and Bačak, and similarly for CAT(1)-space is true that Mosco convergence implies convergence of resolvents, which in turn imply convergence of gradient flows on to the CAT(1) setting. The techniques employ weak convergence in CAT(1)-spaces and cover asymptotic relations of sequences of such spaces introduced by Kuwae-Shioya, including Gromov-Hausdorff limits. The Mosco convergence of non-negative convex lower semi-continuous functions results pointwise convergence of Moreau envelopes in a CAT(0)-space [2]. The related inverse implication was set up by Bačak, Montag and Steidl in CAT(0)-spaces, which extension is represents a new research area on the CAT(1)-spaces. The paper demonstrates a numerical example of Douglas-Rachford operator splitting algorithm, using the Moreau-Yosida envelop theorem.

Keywords

Status: accepted

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