9 research outputs found
Convergence of nonlinear semigroups under nonpositive curvature
The present paper is devoted to semigroups of nonexpansive mappings on metric
spaces of nonpositive curvature. We show that the Mosco convergence of a
sequence of convex lsc functions implies convergence of the corresponding
resolvents and convergence of the gradient flow semigroups. This extends the
classical results of Attouch, Brezis and Pazy into spaces with no linear
structure. The same method can be further used to show the convergence of
semigroups on a sequence of spaces, which solves a problem of [Kuwae and
Shioya, Trans. Amer. Math. Soc., 2008].Comment: Accepted for publication in Trans. Amer. Math. So
A new proof of the Lie-Trotter-Kato formula in Hadamard spaces
The Lie-Trotter-Kato product formula has been recently extended into Hadamard
spaces by [Stojkovic, Adv. Calc. Var., 2012]. The aim of our short note is to
give a simpler proof relying upon weak convergence instead of an ultrapower
technique.Comment: arXiv admin note: text overlap with arXiv:1211.041
Computing medians and means in Hadamard spaces
The geometric median as well as the Frechet mean of points in an Hadamard
space are important in both theory and applications. Surprisingly, no
algorithms for their computation are hitherto known. To address this issue, we
use a split version of the proximal point algorithm for minimizing a sum of
convex functions and prove that this algorithm produces a sequence converging
to a minimizer of the objective function, which extends a recent result of D.
Bertsekas (2001) into Hadamard spaces. The method is quite robust and not only
does it yield algorithms for the median and the mean, but it also applies to
various other optimization problems. We moreover show that another algorithm
for computing the Frechet mean can be derived from the law of large numbers due
to K.-T. Sturm (2002). In applications, computing medians and means is probably
most needed in tree space, which is an instance of an Hadamard space, invented
by Billera, Holmes, and Vogtmann (2001) as a tool for averaging phylogenetic
trees. It turns out, however, that it can be also used to model numerous other
tree-like structures. Since there now exists a polynomial-time algorithm for
computing geodesics in tree space due to M. Owen and S. Provan (2011), we
obtain efficient algorithms for computing medians and means, which can be
directly used in practice.Comment: Corrected version. Accepted in SIAM Journal on Optimizatio
The asymptotic behavior of a class of nonlinear semigroups in Hadamard spaces
We study a nonlinear semigroup associated to a nonexpansive mapping on a
Hadamard space and establish its weak convergence to a fixed point. A
discrete-time counterpart of such a semigroup, the proximal point algorithm,
turns out to have the same asymptotic behavior. This complements several
results in the literature -- both classical and more recent ones. As an
application, we obtain a new approach to heat flows in singular spaces for
discrete, as well as continuous times.Comment: Accepted in JFPT
On proximal mappings with Young functions in uniformly convex Banach spaces
It is well known in convex analysis that proximal mappings on Hilbert spaces
are -Lipschitz. In the present paper we show that proximal mappings on
uniformly convex Banach spaces are uniformly continuous on bounded sets.
Moreover, we introduce a new general proximal mapping whose regularization term
is given as a composition of a Young function and the norm, and formulate our
results at this level of generality. It is our aim to obtain the corresponding
modulus of uniform continuity explicitly in terms of a modulus of uniform
convexity of the norm and of moduli witnessing properties of the Young
function. We also derive several quantitative results on uniform convexity,
which may be of interest on their own.Comment: Accepted in J. Convex Ana
Convex analysis and optimization in Hadamard spaces
This book gives a first systematic account on the subject of convex analysis and optimization in Hadamard spaces. It is primarily aimed at both graduate students and researchers in analysis and optimization.</html