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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)

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)

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)

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