ﻻ يوجد ملخص باللغة العربية
We establish the following result: if the graph of a (nonsmooth) real-extended-valued function $f:mathbb{R}^{n}to mathbb{R}cup{+infty}$ is closed and admits a Whitney stratification, then the norm of the gradient of $f$ at $xin{dom}f$ relative to the stratum containing $x$ bounds from below all norms of Clarke subgradients of $f$ at $x$. As a consequence, we obtain some Morse-Sard type theorems as well as a nonsmooth Kurdyka-L ojasiewicz inequality for functions definable in an arbitrary o-minimal structure.
Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain
We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions.
We provide a characterization of the set of real-valued functions that can be the value function of some polynomial game. Specifically, we prove that a function $u : dR to dR$ is the value function of some polynomial game if and only if $u$ is a continuous piecewise rational function.
We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar
A novel derivative-free algorithm, optimization by moving ridge functions (OMoRF), for unconstrained and bound-constrained optimization is presented. This algorithm couples trust region methodologies with output-based dimension reduction to accelerat