##
**Optimization.**
*(English)*
Zbl 0688.90034

Handbooks in Operations Research and Management Science, 1. Amsterdam etc.: North-Holland. xiv, 709 p. $ 115.00; Dfl. 295.00 (1989).

[The articles of this volume will not be indexed individually.]

The book under review presents ten state-of-the-art expository articles on the most important topics in optimization, written by nineteen outstanding experts in the field. Hence, the book consists of ten chapters. Each chapter starts at an introductory level suitable for master’s level graduate students, but includes discussion of the references to the latest developments in the area. The primary topics in the book include the following: unconstrained optimization, linear, nonlinear, integer, non-differentiable, stochastic and global optimization, network flows, polyhedral combinatorics and multiple criteria decision making. A bibliography follows each chapter and contains only works which have influenced the authors, and which the authors consider to be the most important works in this field, including of course their own.

The first chapter “A view of unconstrained optimization”, by J. E. Dennis and R. B. Schnabel, discusses methods for solving unconstrained optimization problems: Newton’s method for nonlinear minimization, quasi-Newton and finite-difference approximations, globally convergent algorithms, non-Taylor series methods (the Nelder-Mead simplex method and conjugate gradient methods) and recent research on large-scale problems, data-fitting applications, secant updates, singular problems and parallel computation.

Chapter II “Linear programming” written by D. Goldfarb and M. J. Todd deals with some classical questions related to linear programming: the simplex method, the geometry of linear programming models, duality theory and sensitivity analysis, applications of the revised simplex method to large and structured problems. The final sections provide detailed descriptions of the ellipsoid method and Karmarkar’s projective method, which are distinguished from the simplex method in that they have the desirable theoretical property of polynomial-time boundedness.

Chapter III “Constrained nonlinear programming” written by P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, discusses constrained and nonlinear programming (equality and inequality constraints). The optimality conditions and the important special case of quadratic programming are presented. The chapter presents a class of methods: sequential quadratic programming methods, sequential linearly constrained methods, augmented Lagrangian methods, and barrier-function methods.

Chapter IV “Network flows” by R. K. Ahuja, T. L. Magnanti and J. B. Orlin summarizes many of the fundamental ideas of network optimization with emphasis on network form problems (the shortest path problem, the maximum flow problem and the minimum cost flow problem).

Chapter V “Polyhedral combinatorics” by W. R. Pulleyblank reports on how to represent the feasible region of a discrete optimization problem by a polyhedron and thus reduce the problem to a linear program. There are sections on the techniques of polyhedral combinatorics, the techniques of separation and polarity, the dimension and adjacency relations of polyhedra.

The following chapter “Integer programming” by G. L. Nemhauser and L. A. Wolsey, continues the development of discrete optimization to encompass more general and computationally difficult problems. The material presented in this chapter is taken almost entirely from their book “Integer and combinatorial optimization” (1988; Zbl 0652.90067).

Chapter VII “Nondifferentiable optimization”, by C. LemarĂ©chal, is on the subject of nondifferentiable optimization, i.e. optimization of a function which fails to have derivatives for some values of the variables. Examples of nonsmooth problems, special methods for special problems, subgradient and bundle methods and directions for future developments are presented.

Stochastic programming is the subject of chapter VIII by R. J.-B. Wets. The chapter provides the mathematical basis for several of the better known solution methods, such as stochastic quasi-gradient methods, scenario aggregation, decomposition and Monte Carlo methods.

Global optimization is the subject of chapter IX, by A. H. G. Rinnooy Kan and G. T. Timmer.

The final chapter, “Multiple criteria decision making: Five basic concepts” by P.L. Yu introduces a variety of approaches from multi-criterion decision making including goal programming, maximum programming, interactive methods, the construction of utility functions, the elimination of all but undominated feasible points and special simplex methods to cover the linear programming subclass.

The reviewer found this handbook very interesting, clearly and professionally written and inspiring as a source for further researches. Everyone who is interested in optimization should be a acquainted with this work.

The book under review presents ten state-of-the-art expository articles on the most important topics in optimization, written by nineteen outstanding experts in the field. Hence, the book consists of ten chapters. Each chapter starts at an introductory level suitable for master’s level graduate students, but includes discussion of the references to the latest developments in the area. The primary topics in the book include the following: unconstrained optimization, linear, nonlinear, integer, non-differentiable, stochastic and global optimization, network flows, polyhedral combinatorics and multiple criteria decision making. A bibliography follows each chapter and contains only works which have influenced the authors, and which the authors consider to be the most important works in this field, including of course their own.

The first chapter “A view of unconstrained optimization”, by J. E. Dennis and R. B. Schnabel, discusses methods for solving unconstrained optimization problems: Newton’s method for nonlinear minimization, quasi-Newton and finite-difference approximations, globally convergent algorithms, non-Taylor series methods (the Nelder-Mead simplex method and conjugate gradient methods) and recent research on large-scale problems, data-fitting applications, secant updates, singular problems and parallel computation.

Chapter II “Linear programming” written by D. Goldfarb and M. J. Todd deals with some classical questions related to linear programming: the simplex method, the geometry of linear programming models, duality theory and sensitivity analysis, applications of the revised simplex method to large and structured problems. The final sections provide detailed descriptions of the ellipsoid method and Karmarkar’s projective method, which are distinguished from the simplex method in that they have the desirable theoretical property of polynomial-time boundedness.

Chapter III “Constrained nonlinear programming” written by P. E. Gill, W. Murray, M. A. Saunders and M. H. Wright, discusses constrained and nonlinear programming (equality and inequality constraints). The optimality conditions and the important special case of quadratic programming are presented. The chapter presents a class of methods: sequential quadratic programming methods, sequential linearly constrained methods, augmented Lagrangian methods, and barrier-function methods.

Chapter IV “Network flows” by R. K. Ahuja, T. L. Magnanti and J. B. Orlin summarizes many of the fundamental ideas of network optimization with emphasis on network form problems (the shortest path problem, the maximum flow problem and the minimum cost flow problem).

Chapter V “Polyhedral combinatorics” by W. R. Pulleyblank reports on how to represent the feasible region of a discrete optimization problem by a polyhedron and thus reduce the problem to a linear program. There are sections on the techniques of polyhedral combinatorics, the techniques of separation and polarity, the dimension and adjacency relations of polyhedra.

The following chapter “Integer programming” by G. L. Nemhauser and L. A. Wolsey, continues the development of discrete optimization to encompass more general and computationally difficult problems. The material presented in this chapter is taken almost entirely from their book “Integer and combinatorial optimization” (1988; Zbl 0652.90067).

Chapter VII “Nondifferentiable optimization”, by C. LemarĂ©chal, is on the subject of nondifferentiable optimization, i.e. optimization of a function which fails to have derivatives for some values of the variables. Examples of nonsmooth problems, special methods for special problems, subgradient and bundle methods and directions for future developments are presented.

Stochastic programming is the subject of chapter VIII by R. J.-B. Wets. The chapter provides the mathematical basis for several of the better known solution methods, such as stochastic quasi-gradient methods, scenario aggregation, decomposition and Monte Carlo methods.

Global optimization is the subject of chapter IX, by A. H. G. Rinnooy Kan and G. T. Timmer.

The final chapter, “Multiple criteria decision making: Five basic concepts” by P.L. Yu introduces a variety of approaches from multi-criterion decision making including goal programming, maximum programming, interactive methods, the construction of utility functions, the elimination of all but undominated feasible points and special simplex methods to cover the linear programming subclass.

The reviewer found this handbook very interesting, clearly and professionally written and inspiring as a source for further researches. Everyone who is interested in optimization should be a acquainted with this work.

Reviewer: I.M.Stancu-Minasian

### MSC:

90Cxx | Mathematical programming |

00Bxx | Conference proceedings and collections of articles |

52Bxx | Polytopes and polyhedra |

90B50 | Management decision making, including multiple objectives |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

90B10 | Deterministic network models in operations research |

65K05 | Numerical mathematical programming methods |