We discuss new applications of advanced tools of variational analysis and generalized differentiation to a number of important problems in optimization theory, equilibria, optimal control, and feedback control design. The presented results are largely based on the recent work by the author and his collaborators. Among the main topics considered and briefly surveyed in this paper are new calculu...

(1) x0 ∈ K such that c1 ≤ f(x0, y) ≤ c2, for all y ∈ K, where K is a given set, f : K × K → R is a given function and c1 and c2 two real numbers such that c1 ≤ c2. If K is a nonempty closed subset in a locally convex semi-reflexive topological vector space, then the problem (1) is problem by Isac, Sehgal and Singh [16] (Open Problem 2). In [20], Li gave the answer of this problem by introducing...

Bogdan and Kolumbán [3] gave sufficient conditions for closedness of the solution map defined on the set of parameters. They considered the parametric equilibrium problems governed by topological pseudomonotone maps depending on a parameter. In this paper we generalize this result for parametric vector equilibrium problems with trifunctions. Let X and Y be Hausdorff topological spaces and P , t...

In this paper, we study some growth properties of entire functions represented by a vector valued Dirichlet series on the basis of generalized relative Ritt order and generalized relative Ritt lower order.

In this article, we introduce a perturbed system of generalized mixed quasi-equilibrium-like problems involving multi-valued mappings in Hilbert spaces. To calculate the approximate solutions of the perturbed system of generalized multi-valued mixed quasi-equilibrium-like problems, firstly we develop a perturbed system of auxiliary generalized multi-valued mixed quasi-equilibrium-like problems,...

This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type AN−1. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several objects and properties are analyzed, such as the canonical bilinear form which pairs polynomials with those arising from reciprocals of the original parameters, and t...

We consider the general problem of learning an unknown functional dependency, f : X !→ Y, between a structured input space X and a structured output space Y, from labeled and unlabeled examples. We formulate this problem in terms of data-dependent regularization in Vector-valued Reproducing Kernel Hilbert Spaces (Micchelli & Pontil, 2005) which elegantly extend familiar scalarvalued kernel meth...

The elicitation of a statistic, or property of a distribution, is the task of devising proper scoring rules, equivalently proper losses, which incentivize an agent or algorithm to truthfully estimate the desired property of the underlying probability distribution or data set. Leveraging connections between elicitation and convex analysis, we address the vector-valued property case, which has re...

Quasi-complete, locally convex topological vector spaces V have the useful property that continuous compactly-supported V -valued functions have integrals with respect to finite Borel measures. Rather than constructing integrals as limits following [Bochner 1935], [Birkhoff 1935], et alia, we use the [Gelfand 1936][Pettis 1938] characterization of integrals, which has good functorial properties...