728 research outputs found
On the existence of a solution to a spectral estimation problem \emph{\`a la} Byrnes-Georgiou-Lindquist
A parametric spectral estimation problem in the style of Byrnes, Georgiou,
and Lindquist was posed in \cite{FPZ-10}, but the existence of a solution was
only proved in a special case. Based on their results, we show that a solution
indeed exists given an arbitrary matrix-valued prior density. The main tool in
our proof is the topological degree theory.Comment: 6 pages of two-column draft, accepted for publication in IEEE-TA
Well-posedness in vector optimization and scalarization results
In this paper, we give a survey on well-posedness notions of Tykhonov's type for vector optimization problems and the links between them with respect to the classification proposed by Miglierina, Molho and Rocca in [33]. We consider also the notions of extended well-posedness introduced by X.X. Huang ([19],[20]) in the nonparametric case to complete the hierchical structure characterizing these concepts. Finally we propose a review of some theoretical results in vector optimization mainly related to different notions of scalarizing functions, linear and nonlinear, introduced in the last decades, to simplify the study of various well-posedness properties.
The Radius of Metric Subregularity
There is a basic paradigm, called here the radius of well-posedness, which
quantifies the "distance" from a given well-posed problem to the set of
ill-posed problems of the same kind. In variational analysis, well-posedness is
often understood as a regularity property, which is usually employed to measure
the effect of perturbations and approximations of a problem on its solutions.
In this paper we focus on evaluating the radius of the property of metric
subregularity which, in contrast to its siblings, metric regularity, strong
regularity and strong subregularity, exhibits a more complicated behavior under
various perturbations. We consider three kinds of perturbations: by Lipschitz
continuous functions, by semismooth functions, and by smooth functions,
obtaining different expressions/bounds for the radius of subregularity, which
involve generalized derivatives of set-valued mappings. We also obtain
different expressions when using either Frobenius or Euclidean norm to measure
the radius. As an application, we evaluate the radius of subregularity of a
general constraint system. Examples illustrate the theoretical findings.Comment: 20 page
Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions
This paper is devoted to the study of generalized differentiation properties
of the infimal convolution. This class of functions covers a large spectrum of
nonsmooth functions well known in the literature. The subdifferential formulas
obtained unify several known results and allow us to characterize the
differentiability of the infimal convolution which plays an important role in
variational analysis and optimization
On the well-posedness of multivariate spectrum approximation and convergence of high-resolution spectral estimators
In this paper, we establish the well-posedness of the generalized moment
problems recently studied by Byrnes-Georgiou-Lindquist and coworkers, and by
Ferrante-Pavon-Ramponi. We then apply these continuity results to prove almost
sure convergence of a sequence of high-resolution spectral estimators indexed
by the sample size
Well-Behavior, Well-Posedness and Nonsmooth Analysis
AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter
notion means that any critical sequence (xn) of a lower semicontinuous function
f on a Banach space is minimizing. Here âcriticalâ means that the remoteness of
the subdifferential âf(xn) of f at xn (i.e. the distance of 0 to âf(xn)) converges
to 0. The objective function f is not supposed to be convex or smooth and the
subdifferential â is not necessarily the usual Fenchel subdifferential. We are thus
led to deal with conditions ensuring that a growth property of the subdifferential
(or the derivative) of a function implies a growth property of the function itself.
Both qualitative questions and quantitative results are considered
Advances in Optimization and Nonlinear Analysis
The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics
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