Mathematical vector optimization in partially ordered linear spaces. (English) Zbl 0578.90048

Methoden und Verfahren der Mathematischen Physik, Bd. 31. Frankfurt am Main-Bern-New York: Verlag Peter Lang. IX, 310 p. SFr 65.00 (1986).
As the title indicates, this text is concerned with the mathematical foundations of vector optimization in general partially ordered linear vector spaces, including, as special cases, partially ordered Hilbert and Banach spaces. Chapters 5 to 8 are the main chapters dealing largely with concepts of minimal, strongly minimal, properly minimal and weakly minimal elements of a set \(S\subseteq X\), when the partial order \(\leq\) over a set X is with respect to some cone \(C_ X\) and defined by \(x\leq y\rightleftarrows y-x\in C_ X\). More common terms for these elements are efficient solutions, least elements, properly efficient solutions, and weakly efficient solutions. Chapter 5 deals with necessary and sufficient conditions for elements to be in the specified categories in terms of the existence of norms, seminorms, or linear-functionals and monotonic functionals which are minimised at these elements. Chapter 6 deals with conditions for the existence of elements of the kinds specified. Chapter 7, making use of Fréchet differentiability, considers the case, when Banach space X is mapped by a function f (referred to as the objective mapping) into a partially ordered normed space Y, and suggested to constraints, in cone terms. An extension of the Fritz-John, and Kuhn-Tucker optimality conditions, for scalar optimization problems is given for elements to be weakly minimal or locally minimal. Chapter 8 deals with duality results for the vector optimization problems.
The text is well supported with some basic material in analysis, covering linear spaces, cones, partial orders, convex analysis, differentiable mapping, and such things as Zorn’s lemma, in Chapters 1 to 3, some introductory material on optimality notions in Chapter 4, and is rounded off with Chapters 9 and 10, applying the earlier results to vector approximation and to the derivation of a maximum principle for cooperative n-person differential games.
The text, although to some extent fairly self contained, will demand a considerable amount of mathematical maturity from the reader. It will be indispensible to those interested in the mathematical foundations of vector optimization.
Reviewer: D.J.White


90C31 Sensitivity, stability, parametric optimization
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
91A23 Differential games (aspects of game theory)
46A40 Ordered topological linear spaces, vector lattices
91A12 Cooperative games