Integral global optimization. Theory, implementation and applications
type: Item
instance of: scholarly article
title: Integral global optimization. Theory, implementation and applications
zbMATH Open document ID: 0689.90052
author: Soo Hong Chew
author: Quan Zheng
published in: Lecture Notes in Economics and Mathematical Systems
publication date: 1993-06-05T00:00:00Z
review text: The book is divided into five chapters. The introductory Chapter I formulates first the problem to be studied: Let X be a Hausdorff space, f a real valued function on X, and S a closed subset of X. The problem is to find \(\bar c=\inf_{x\in S}f(x)\). It is assumed that f is continuous on S and that there exists a real number c such that the intersection of S and the level set \(H_ c=\{x|\) f(x)\(\leq c\}\) is nonempty and compact. let (X,\(\Omega\),\(\mu)\) be a measure space such that (i) \(\Omega\) is a Borel \(\sigma\)-field (i.e., \(\Omega\) contains each open subset of X), (ii) every nonempty open subset of X has a positive measure, (iii) the measure of every compact set is finite. If (X,\(\Omega\),\(\mu)\) has these properties it is called a Q- measure space and \(\mu\) is called a Q-measure. A subset G of X is called robust if cl(int G)\(=cl G.\) Chapter II is entitled ``Integral characterizations of global optimality''. If \(c>\bar c=\min f(x)\) we define the mean value M(f,c) of f over its level set \(H_ c\) by the equation \[ M(f,c)=\frac{1}{\mu (H_ c)}\int_{H_ c}f(x)d\mu, \] where \(\mu\) is a Q-measure. This definition is extended to cover the situation when \(\mu (H_ c)\) may vanish \((c=\bar c)\) by the limit definition: Let \(c\geq \bar c\) and let \(\{c_ k\}\) be a decreasing sequence of real numbers converging to c. Then \(M(f,c)=\lim_{c_ k\downarrow c}M(f,c_ k)\). A number of properties of M(f,c) are derived; in particular, it is shown that the following three statements are equivalent: a) \(\bar x\) is a global minimum of f(x) with \(f(\bar x)=\bar c\). b) \(M(f,c)\geq \bar c\) for \(c>\bar c\). c) \(M(f,\bar c)=\bar c\). The concept of the mean value is then extended to that of the variance and higher moments of f over its level sets. Using these moments, a number of optimality conditions are proved. In Section 3, these results are applied to certain constrained cases. Section 4 deals with a modification of the penalty method and Section 5 is dev…
reviewed by: Jaromir Abrham
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 90-02
Mathematics Subject Classification ID: 90C10
Mathematics Subject Classification ID: 90C11
zbMATH DE Number: 193173
zbMATH Keywords: integral global optimization
zbMATH Keywords: Hausdorff space
zbMATH Keywords: Q-measure
zbMATH Keywords: functions with discontinuities
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description: scientific article; zbMATH DE number 193173