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Contents Preface ix Chapter 1. GENERAL OPTIMALITY 1.1. $-subgradients and $-supergradients 1 1.2. Duality 16 1.3. Optimization problems with constraints 21 1.4. ^-convex sets 23 1.5. ^-convexity in linear spaces 28 1.6. ^-separation 37 1.7. Constraints of multifunction type 40 1.8. Polarity and duality 50 Chapter 2. OPTIMIZATION IN METRIC SPACES 2.1.… (More)

- S. ROLEWICZ
- 2005

In the theory of optimization an essential role is played by the differentiability of convex functions. In this paper we shall try to extend the results concerning differentiability to a larger class of functions called strongly α(·)-paraconvex. Let (X, .) be a real Banach space. Let f (x) be a real valued strongly α(·)-paraconvex function defined on an… (More)

- S. Rolewicz
- 2007

In 1933 S. Mazur [4] proved the following Theorem 1. Let (X, ·) be a separable real Banach space. Let f be a real-valued convex continuous function defined on an open convex subset Ω ⊂ X. Then there is a subset A ⊂ Ω of the first category such that f is Gateaux differentiable on Ω \ A. The result of Mazur was a starting point for the theory of… (More)

- D N Kutzarova, S Rolewicz
- 2007

Let (X, ·) be a real Banach space. Let C be a closed convex set in X. By a drop D(x, C) determined by a point x ∈ X, x / ∈ C, we shall mean the convex hull of the set {x} ∪ C. We say that C has the drop property if C = X and if for every nonvoid closed set A disjoint with C, there exists a point a ∈ A such that D(a, C) ∩ A = {a}. For a given C a sequence {x… (More)

- Szczepan Perz, Stefan Rolewicz
- ZOR - Meth. & Mod. of OR
- 1990

It is shown that in metric spaces each (α, φ)-meagre set A is uniformly very porous and its index of uniform v-porosity is not smaller than k−α 3k+α , provided that φ is a strictly k-monotone family of Lipschitz functions and α < k. The paper contains also conditions implying that a k-monotone family of Lipschitz functions is strictly k-monotone. Let (X, d)… (More)

In the paper a class of families F(M) of functions defined on dif-ferentiable manifolds M with the following properties:

In this paper a concept of a generalized directional derivative, which satisfies Leibniz rule is proposed for locally Lip-schitz functions, defined on an open subset of a Banach space. Although Leibniz rule is of less importance for a subdifferential calculus , it is of course of some theoretical interest to know about the existence of generalized… (More)

- D N Kutzarova, S Rolewicz
- 2007

The notion of nearly uniformly convex Banach spaces was introduced by Huff [2] and independently by Goebel and Sekowski [1]. In [7] it was shown that in a certain way it is a uniformization of drop property for norms. In [3] the authors considered drop property for convex sets. In the present paper they investigate nearly uniformly convex sets which need… (More)

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