Optimization by vector space methods. (English) Zbl 0176.12701

Luenberger, D. G.

Series in Decision and Control. New York-London. Sydney-Toronto: John Wiley and Sons, Inc., xiii, 326 p. (1969).

This is a book which treats various aspects of the problems of optimization of functions and functionals. Although the author mentions in his preface that the book is aimed at graduate students among which he lists engineers, it seems to me that it will be best appreciated by pure mathematicians or by others who have a good background in analysis.
In Chapter 1 the reader is introduced, by means of examples, to various problems in which optimization techniques can be employed. Then in Chapter 2 one finds the basic results in vector space theory which are essential in understanding most of the rest of the book. The topics chosen in this and in Chapter 3 are well suited to the applications which are made later.
Chapter 4 deals with some optimization results arising from Probability theory, and Chapter, 5 contains results on Banach spaces, some of which generalize previous conclusions on Hilbert spaces in Chapter 3. Chapter 6 contains further results on Banach spaces which are applied in Chapters 7 and 8 in the treatment of duality principles, the min-max theorem, and Lagrange multipliers. In each of these chapters, and indeed throughout the rest of the book, geometrical analogues of the results are given. Although these prove useful in understanding some of the principles involved and for providing examples, the introduction of the concepts in the proof of theorems tend, in some cases, to make them harder to follow. A good understanding of the basic analytical theory seems essential in these parts of the book.
Chapter 9, which deals with the local theory of constrained optimization contains further topics on Lagrange multipliers and includes the Kuhn-Tucker theorem and a treatment of the Pontryagin maximum principle. The final chapter deals with various special methods including Newton’s method, method of steepest descent and other processes involving successive approximation.
The book contains a good selection of references and bibliography, and each chapter ends with a set of problems and a list of references for each section. Although most of the solved examples would prove useful to the analyst, one who is mainly interested in applications may have to refer to some of the titles given. The book is well-written and the typography is alright.
There are, however, a number of misprints; one of which is mathematical in nature occurs on P. 51 line 9, where one of the expressions should read \(\Vert x - m_0\Vert = \delta\). One main shortcoming I noticed is the failure in some instances to give adequate references (mainly involving previous statements and theorems). A notable case of this is the failure to mention that the proofs of Propositions 1 and 2 section 6.2 are similar to those of Propositions 1 and 2 in section 5.2. Another instance occurs in cases where \(L_p\)-spaces are mentioned. This seems to be due to the exclusion of some of the results involving these spaces. Despite these faults, as can be seen from my previous comments, the author has provided a readable book on optimization theory by vector methods.

Reviewer: George O. Okikiolu (London)

MSC:

90-02	Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C48	Programming in abstract spaces
65K05	Numerical mathematical programming methods

Keywords:

duality principles; min-max theorem; Lagrange multipliers; Kuhn-Tucker theorem; Pontryagin maximum principle; Newton’s method; method of steepest descent; successive approximation.