zbMATH — the first resource for mathematics

Nonlinear and mixed-integer optimization. Fundamentals and applications. (English) Zbl 0886.90106
Oxford: Oxford Univ. Press. xv, 462 p. (1995).
The book under review is a thorough treatise on nonlinear mixed-integer optimization and seems to be the first general book on this subject so far as the fundamentals and applications are concerned. It sets a standard for clarity and completeness that will be difficult to surpass. The author presents a comprehensive survey of this field, to which he himself has notably contributed, and thus combines and unifies diverse results that hitherto were scattered throughout the literature.
Nonlinear and mixed-integer optimization address the problem of optimization address the problem of optimizing an objective function subject to equality and inequality constraints in the presence of continuous and integer variables. These optimization models have many applications in engineering and applied science problems and this is the primary motivation for the theoretical and algorithmic developments that researchers have been experiencing during the last two decades.
The author mainly deals with the fundamental theoretical aspects and the algorithmic issues of nonlinear mixed-integer optimization models and discusses their applications in the important area of process synthesis in chemical engineering. The author writes with a consistent point of view and a style which is very readable, flowing smoothly from topic to topic. The book is not only great as a whole, it also seems perfect in every detail. There are historical notes and a very extensive bibliography. The book has three main parts. The first part dealing with general abstract theory of fundamentals of convex analysis and nonlinear optimization, has four chapters. It covers convex sets, generalization of convex and concave functions, unconstrained and constrained nonlinear optimization. Nonlinear optimization problem has two different representations, the primal problem and the dual problem in relation between the primal and dual problem is provided by an elegant duality theory. The concept of duality theory plays a leading role and exploits methods of convex analysis culminating in a perturbation function associated with the primal problem that establishes the relationship between the existence of optimal multipliers and the stability of the primal problem. The presentation of weak and strong duality theorems, duality gap, connection between the continuity of the perturbation function and the existence of the duality gap enriches the theory.
The second part, having two chapters, addresses the fundamentals and algorithms for mixed-integer linear and nonlinear optimization models. There are a brief introduction to the basic notions of a branch and bound algorithmic framework, description of a general branch and bound algorithm and a linear relaxation-based branch and bound approach; these ideas are illustrated with a simple example. These materials are intended only as a basic introduction to mixed-integer linear programming problems. These mixed-integer linear programming problems are employed as subproblems in the mixed-integer nonlinear programming approaches.
The third part, with four chapters, initially introduces the generic problems in the area of process synthesis. Process synthesis an important research area within chemical process design, has triggered during the last three decades a significant amount of academic research work and industrial interest. This part also discusses key ideas in the mathematical modeling of process system, and concentrates on the important application areas of heat exchanger networks, separation system synthesis, and reactor-based system synthesis. Extensive reviews exist for the process synthesis area as well as for special classes of problems and for particular approaches applied to process synthesis problems.
The book can be used for a graduate course in optimization and process synthesis; it would certainly be a wonderful supplement to a standard text. Because it comprehensively reviews the literature which is widely scattering throughout a great variety of journal articles, this book will be also a valuable addition to the library of any researcher seriously interested in this field. This book is outstanding in its choice of materials. It offers the reader an opportunity to progress from minimal prerequisites to substantial competence in optimization and process synthesis; the book is an excellent introduction to this beautiful and difficult subject.

90C11 Mixed integer programming
90-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming
90C30 Nonlinear programming