170 research outputs found
An Implicit-Function Theorem for B-Differentiable Functions
A function from one normed linear space to another is said to be Bouligand differentiable (B-differentiable) at a point if it is directionally differentiable there in every direction, and if the directional derivative has a certain uniformity property. This is a weakening of the classical idea of Frechet (F-) differentiability, and it is useful in dealing with optimization problems and in other situations in which F-differentiability may be too strong.
In this paper we introduce a concept of strong B-derivative, and we employ this idea to prove an implicit-function theorem for B-differentiable functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with B-differentiability in place of F-differentiability. Therefore it is applicable to a considerably wider class of functions than is the classical theorem
Relative Well-Posedness of Constrained Systems with Applications to Variational Inequalities
The paper concerns foundations of sensitivity and stability analysis, being
primarily addressed constrained systems. We consider general models, which are
described by multifunctions between Banach spaces and concentrate on
characterizing their well-posedness properties that revolve around Lipschitz
stability and metric regularity relative to sets. The enhanced relative
well-posedness concepts allow us, in contrast to their standard counterparts,
encompassing various classes of constrained systems. Invoking tools of
variational analysis and generalized differentiation, we introduce new robust
notions of relative coderivatives. The novel machinery of variational analysis
leads us to establishing complete characterizations of the relative
well-posedness properties with further applications to stability of affine
variational inequalities. Most of the obtained results valid in general
infinite-dimensional settings are also new in finite dimensions.Comment: 25 page
Stability analysis for parameterized variational systems with implicit constraints
In the paper we provide new conditions ensuring the isolated calmness
property and the Aubin property of parameterized variational systems with
constraints depending, apart from the parameter, also on the solution itself.
Such systems include, e.g., quasi-variational inequalities and implicit
complementarity problems. Concerning the Aubin property, possible restrictions
imposed on the parameter are also admitted. Throughout the paper, tools from
the directional limiting generalized differential calculus are employed
enabling us to impose only rather weak (non-restrictive) qualification
conditions. Despite the very general problem setting, the resulting conditions
are workable as documented by some academic examplesComment: 26 page
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 Delta Theorems for Multivalued Mappings and Measurable Selections
The classical delta theorem can be generalized in a mathematically satisfying way to a broad class of multivalued and/or nonsmooth mappings, by examining the convergence in distribution of the sequence of difference quotients from the perspectives of recent developments in convergence theory for random closed sets and new descriptions of first-order behavior of multivalued mappings. Such a theory opens the way to applications of asymptotic techniques in many areas of mathematical optimization where randomness and uncertainty play a role. Of special importance is the asymptotic convergence of measurable selections of multifunctions when the limit multifunction is single-valued almost surely
On the isolated calmness property of implicitly defined multifunctions
The paper deals with an extension of the available theory of SCD (subspace
containing derivatives) mappings to mappings between spaces of different
dimensions. This extension enables us to derive workable sufficient conditions
for the isolated calmness of implicitly defined multifunctions around given
reference points. This stability property differs substantially from isolated
calmness at a point and, possibly in conjunction with the Aubin property,
offers a new useful stability concept. The application area includes a broad
class of parameterized generalized equations, where the respective conditions
ensure a rather strong type of Lipschitztan behavior of their solution maps.Comment: arXiv admin note: text overlap with arXiv:2106.0051
- …