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2006 | Book

Preliminaries Read chapter Read first chapter

Duality for Nonconvex Approximation and Optimization

Author: Ivan Singer

Publisher: Springer New York

Book Series : CMS Books in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Table of Contents

Frontmatter
1. Preliminaries
Abstract
In this section we recall some basic definitions and results about convex analysis in the framework of normed linear spaces, in which we shall study the approximation problems, and in the more general framework of locally convex spaces in which we shall study the optimization problems.
2. Worst Approximation
Abstract
We recall that the deviation (or excess) of a set G (assumed nonempty in this chapter, without any special mention) from an element XQ in a normed linear space X is the number δ(G, χ0) ≥ 0 defined by
$$ \delta (G,x_0 ): = \mathop {sup}\limits_{g \in G} \parallel g - x_0 \parallel , $$
(2.1)
and any g0∈G for which this sup is attained, i.e., such that
$$ \parallel g_0 - x_0 \parallel = \mathop {sup}\limits_{g \in G} \parallel g - x_0 \parallel , $$
(2.2)
or equivalently, such that
$$ \parallel g_0 - x_0 \parallel \geqslant \parallel g - x_0 \parallel (g \in G), $$
(2.3)
is called an element of worst approximation of (or a farthest point to) x0in G (see Figure 2.1).
3. Duality for Quasi-convex Supremization
Abstract
Given a locally convex space X with conjugate space X*, a subset G of X and a quasi-convex function ƒ : X → \( \bar R \), in this chapter we shall give duality results for the primal supremization problem
$$ (P^s ) = (P_{G,f}^s ) \alpha ^s = \alpha _{G,f}^s = \sup f(G). $$
(3.1)
4. Optimal Solutions for Quasi-convex Maximization
Abstract
Let X be a locally convex space, ƒ : X ƒ; \( \bar R \) a function, G ⊆ Z, and go ∈ G. Clearly, if ƒ(g0) = +∞ then g0 is an optimal solution of the primal supremization problem(Ps](of(3.1)),i.e.,ƒ(g0) = max ƒ(G),andifƒ(go) = -∞, ƒ|G # -∞, then go is not a maximum point of → on G
5. Reverse Convex Best Approximation
Abstract
The study of reverse convex best approximation, that is, of best approximation by complements of convex sets, is motivated, among others, by its connections with the famous unsolved problem whether in a Hilbert space every Chebyshev set (i.e., such that each x ∈ X has a unique element of best approximation in the set) is necessarily convex.
6. Unperturbational Duality for Reverse Convex Infimization
Abstract
Given a locally convex space X, with conjugate space X*, a convex subset G of X, and a function ƒ: X → \( \bar R \), in this chapter we shall give some results of unperturbational duality for the primal “reverse convex infimization” problem
$$ (P^r ) = (P_{G,f}^r ) \alpha ^r = \alpha _{G,f}^r = \inf f(CG). $$
(6.1)
7. Optimal Solutions for Reverse Convex Infimization
Abstract
Let X be a set, → : X → \( \bar R \) a function, G ⊆ X, and z0 ⊆ CG. Clearly, if →(z0) = ∞, then z0 is an optimal solution of the primal infimization problem (Pr) (of 6.1), i.e., →(z0) = min/(CG), and if →(z0) = +∞, →|CG ≢ +∞, then z0 is not a minimum point of → on CG. Therefore, the cases of interest are those where
$$ f(z_0 ) \in R. $$
(7.1)
8. Duality for D.C. Optimization Problems
Abstract
In the preceding chapters we have considered optimization problems with various primal constraint sets and objective functions. In this chapter we shall focus on “d.c. optimization problems,” i.e., optimization problems involving differences of convex functions.
9. Duality for Optimization in the Framework of Abstract Convexity
Abstract
In the preceding chapters we have studied duaUty for nonconvex optimization in locally convex spaces, using convexity of sets and/or convexity or quasi-convexity of functions, and we have also given some scattered duality results for optimization, in the framework of abstract convexity, for example. Theorems 3.11, 6.10, and 8.11 on Δ′Δ-convex sets and Δ′Δ-quasi-convex functions.
10. Notes and Remarks
Abstract
A brief list of some standard books on normed linear spaces and locally convex spaces has been mentioned in Chapter 1, Remark 1.1 (a).
From the literature on separation theorems let us also recall that a function Φ ∈ X*\{0} is said to strongly separate C1 from C2 if sup Φ(C1) < inf Φ(C2); clearly, in this case Φ also strictly separates C1 from C2, but in general, the converse is not true.
Backmatter
Metadata
Title
Duality for Nonconvex Approximation and Optimization
Author
Ivan Singer
Copyright Year
2006
Publisher
Springer New York
Electronic ISBN
978-0-387-28395-1
Print ISBN
978-0-387-28394-4
DOI
https://doi.org/10.1007/0-387-28395-1

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