Integral global optimization. Theory, implementation and applications
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 devoted to convex programming problems. The remaining three sections of Chapter II explore the relation between the above approach and the Kuhn-Tucker conditions, the integer and mixed-integer programming, and optimization of certain functions with discontinuities. Chapter III deals with theoretical algorithms developed on the basis of previous results. Chapter IV is devoted to computer implementation of these algorithms and Chapter V (Applications) discusses a number of examples on implementation in terms of the specific methods used.
scientific article; zbMATH DE number 193173