Journal of Convex Analysis 24 (2017), No. 2, 679--684
Copyright Heldermann Verlag 2017
A Note on the Paper "Proximal Point Methods for Quasiconvex and Convex Functions with Bregman Distances on Hadamard Manifolds"
João Xavier Cruz Neto
Dept. of Mathematics, Federal University of Piauí, Teresina, Brazil
jxavier@ufpi.edu.br
I. D. Melo
Dept. of Mathematics, Federal University of Piauí, Teresina, Brazil
P.A. Sousa
Dept. of Mathematics, Federal University of Piauí, Teresina, Brazil
J. P. Silva
Dept. of Mathematics, Federal University of Piauí, Teresina, Brazil
We provide an erratum to Proposition 3.4 of E. A. Papa Quiroz and P. R. Oliveira in their article "Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds" [J. Convex Analysis 16(1) (2009) 49--69]. More specifically, we prove that the function associated with the exponential map in this proposition is not convex on Hadamard manifolds with negative sectional curvature, and present an alternative proof for the result in the case of null sectional curvature.
Keywords: Proximal point algorithm, Hadamard manifold, convex funtion, quasiconvex function, Bregman distance, sectional curvature.
MSC: 65K05, 47J25,90C33,91E10
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