We provide an empirical process theory for locally stationary processes over nonsmooth function classes. An important novelty other approaches is the use of flexible functional dependence measure to quantify dependence. A central limit theorem and nonasymptotic maximal inequalities are provided. The used prove convergence distribution (EDF) derive uniform rates kernel density estimators both pr...

We discuss in this paper a class of nonsmooth functions which can be represented, in a neighborhood of a considered point, as a composition of a positively homogeneous convex function and a smooth mapping which maps the considered point into the null vector. We argue that this is a sufficiently rich class of functions and that such functions have various properties useful for purposes of optimi...

A whole spectrum of subdifferentiability properties is delineated in which various degrees of uniformity are present. Related properties are introduced for sets. Some characterizations in terms of monotonicity properties are displayed. Mathematics Subject Classification: 49J52, 46N10, 46T20.

In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1 )-simplices to generate nodal values for the interpolatio...

A simultaneous perturbation stochastic approximation (SPSA) method has been developed in this paper, using the operators of perturbation with the Lipschitz density function. This model enables us to use the approximation of the objective function by twice differentiable functions and to present their gradients by volume integrals. The calculus of the stochastic gradient by means of this present...

This paper generalizes the notion of risk aversion for functions which are not necessarily differentiable nor strictly concave. Using an approach based on superdifferentials, we define the notion of a risk aversion measure, from which the classical absolute as well as relative risk aversion follows as a RadonNikodym derivative if it exists. Using this notion, we are able to compare risk aversio...

Nonsmoothness pervades optimization, but the way it typically arises is highly structured. Nonsmooth behavior of an objective function is usually associated, locally, with an active manifold: on this manifold the function is smooth, whereas in normal directions it is “veeshaped.” Active set ideas in optimization depend heavily on this structure. Important examples of such functions include the ...

A real valued function g(x, t) on Rn×R is called Lorentz invariant if g(x, t) = g(Ux, t) for all n×n orthogonal matrices U and all (x, t) in the domain of g. In other words, g is invariant under the linear orthogonal transformations preserving the Lorentz cone: {(x, t) ∈ Rn × R | t ≥ ‖x‖}. It is easy to see that every Lorentz invariant function can be decomposed as g = f ◦ β, where f : R2 → R i...