Existence Of Directional Derivatives Research Articles

On directional derivatives for cone-convex functions

We investigate the relationship between the existence of directional derivatives for cone-convex functions with values in a Banach space $Y$ and isomorphisms between $Y$ and $c_0$,

On directional derivatives of selections of set-valued mappings

The paper is devoted to the question of existence of directional derivatives of selections of some set-valued mappings with starlike graphs.

Sensitivity of the limit shape of sample clouds from meta densities

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738-1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos's theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.

On smoothness properties of spatial processes

For inferential analysis of spatial data, probability modelling in the form of a spatial stochastic process is often adopted. In the univariate case, a realization of the process is a surface over the region of interest. The specification of the process has implications for the smoothness of process realizations and the existence of directional derivatives. In the context of stationary processes, the work of Kent (Ann. Probab. 17 (1989) 1432) pursues the notion of a.s. continuity while the work of Stein (Interpolation of Spatial Data; Some Theory for Kriging, Springer, New York, 1999) follows the path of mean square continuity (and, more generally, mean square differentiability). Our contribution is to clarify and extend these ideas in various ways. Our presentation is self-contained and not at a deep mathematical level. It will be of primary value to the spatial modeller seeking greater insight into these smoothness issues.

Second order multipliers and marginal function in non linear programs

For finite dimensional Optimization problems, some first and second order constraints qualification condition is proposed which draws the existence of first and second order multipliers and some sufficient condition is given for the existence of first and second order Lagrange multipliers common to several critical directions. Related results are stated concerning the value function of the programs. Under some Holderian stability assumption the existence of directional derivatives is proved and their expression is given by means of the first and second order Lagrange multipliers.

Saddlepoints for convex programming in order vector spaces

In this paper we characterize the existence of saddlepoints for optimization problems in order complete vector spaces by the existence of order automorphisms of vector space, by the existence of subgradients of the associated perturbing function, and by the existence of directional derivatives of the perturbing function, respectively. The statement of this paper is purely algebraic.