Wyniki wyszukiwania - BazTech

1

On Korenblum convex functions

Ereú J., López L., Merentes N.

Commentationes Mathematicae

|

2017

|

Vol 57, No. 1

25--44

EN

We introduce a new class of generalized convex functions called the K-convex functions, based on Korenblum’s concept of k-decreasing functions, where K is an entropy (distortion) function. We study continuity and differentiability properties of these functions, and we discuss a special subclass which is a counterpart of the class of so-called d.c. functions. We characterize this subclass in terms of the space of functions of bounded second k-variation, extending a result of F. Riesz. We also present a formal structural decomposition result for the K-convex functions.

2

Quasidifferentiable Calculus and Minimal Pairs of Compact Convex Sets

Pallaschke D., Urbański R.

Schedae Informaticae

|

2012

|

Vol. 21

107--125

EN

The quasidifferential calculus developed by V.F. Demyanov and A.M. Rubinov provides a complete analogon to the classical calculus of differentiation for a wide class of nonsmooth functions. Although this looks at the first glance as a generalized subgradient calculus for pairs of subdifferentials it turns out that, after a more detailed analysis, the quasidifferential calculus is a kind of Fréchet-differentiation whose gradients are elements of a suitable Minkowski–Rådström–Hörmander space. One aim of the paper is to point out this fact. The main results in this direction are Theorem 1 and Theorem 5. Since the elements of the Minkowski–Rådström–Hörmander space are not uniquely determined, we focus our attention in the second part of the paper to smallest possible representations of quasidifferentials, i.e. to minimal representations. Here the main results are two necessary minimality criteria, which are stated in Theorem 9 and Theorem 11.

3

The equality case in some recent convexity inequalities

Maksa G., Pales Z.

Opuscula Mathematica

|

2011

|

Vol. 31, no. 2

269-277

EN

In this paper, we investigate a functional equation related to some recently introduced and investigated convexity type inequalities.

4

On strict pseudoconvexity

Ivanov V., I.

Journal of Applied Analysis

|

2007

|

Vol. 13, nr 2

183-196

EN

The present paper provides first and second-order characterizations of a radilly lower semicontinuous strictly pseudoconvex function ∫ : X → R defined on a convex set X in the real Euclidean space Rn in twerms of the lower Dini-directional derivative. In particular we obtain connections between the strictly pseudoconvex functions, nonlinear programming problem, Stampacchia variational inequality, and strict Minty variational inequality. We extend to the radially continuous functions the characterization due to Diewert, Avriel, Zang [6]. A new implication appears in our conditions. Connections with other classes of functions are also derived

5

Rotundity, smoothness and duality

Penot J.-P.

Control and Cybernetics

|

2003

|

Vol. 32, no 4

711-733

EN

The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.

6

Kuhn-Tucker type optimality conditions for some class of nonsmooth programming problems

Śmietański M. J.

Control and Cybernetics

|

2003

|

Vol. 32, no 2

361-376

EN

We consider two nonlinear programming problems with nonsmooth functions. The necessary and sufficient first order optimality conditions use the Dini and Clarke derivatives. However, the obtained Kuhn-Tucker conditions have a rather classical form. The sufficient conditions alone are obtained thanks to some properties of generalized convexity and generalized linearity of functions. The necessary and sufficient optimality conditions are given in the Lagrange form.