79 research outputs found
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World
International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'ĂȘtre" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics
A survey on error bounds for lower semicontinuous functions
We survey ancient and recent results on global error bounds
for the distance to a sublevel set of a lower semicontinuous function defined on
a complete metric space. We emphasize the case of
a recent characterization of this property which appeared in [CITE].
We also review the convex case and show how the known
result on sufficient condition for a global error bound can be derived
from the quoted characterization.
Nonlinear local error bounds via a change of metric
International audienceIn this work, we improve the approach of the second author and V. Motreanu [Math. Program. 114 (2008), 291â319] to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and it allows dealing with local error bounds in an appropriate way. We present some consequences of the general results in the framework of classical nonsmooth analysis, involving Banach spaces and subdifferential operators. In particular, we describe connections between local quadratic growth of a function and metric regularity of its subdifferential
STABILITY OF SUPPORTING AND EXPOSING ELEMENTS OF CONVEX SETS IN BANACH SPACES
Abstract. To a convex set in a Banach space we associate a convex function (the separating function), whose subdifferential provides useful information on the nature of the supporting and exposed points of the convex set. These points are shown to be also connected to the solutions of a minimization problem involving the separating function. We investigate some relevant properties of this function and of its conjugate in the sense of Legendre-Fenchel. Then we highlight the connections between set convergence, with respect to the slice and Attouch-Wets topologies, and convergence, in the same sense, of the associated functions. Finally, by using known results on the behaviour of the subdifferential of a convex function under the former epigraphical perturbations, we are able to derive stability results for the set of supported points and of supporting and exposing functionals of a closed convex subset of a Banach space
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