Nonlinear programming. Theory and algorithms.
3rd ed.

*(English)*Zbl 1140.90040
Hoboken, NJ: John Wiley & Sons (ISBN 0-471-48600-0/hbk; 0-471-78777-9/ebook). xv, 853 p. (2006).

This classical book presents at an undergraduate level a detailed description of the basic theoretical results and algorithms in nonlinear optimization. At some passages this third edition goes actually far beyond the topics usually included into optimization textbooks. The book contains a lot of graphics and examples that support its study. Also very helpful are the exercises at the end of each chapter. They ranges from easy calculations to more theoretical questions. Each chapter includes further a very good bibliography review on the topics treated. The material is organized into three parts: 1.-Convex Analysis; 2.-Optimality conditions and duality; 3.- Algorithms and their convergence. A very first introductory chapter is included before the convex analysis part. It contains interesting examples of nonlinear programming applications in very different disciplines, for instance; optimal control, structural and mechanical design, electrical networks, etc. The part of convex analysis is divided into two chapters (2 and 3). The first one introduce the main concepts and properties of convex sets. The second chapter on this part deals with convex functions including the non differentiable case. The second part is presented into three chapters (4 to 6). The first chapter is devoted to the classical Fritz-John and Karush-Kuhn-Tucker optimality conditions. A section deals also with second order conditions. The second chapter of this part covers extensively the constraint qualifications. In the last chapter 6 the lagrangian duality and the optimality conditions via saddle points are presented. The last part involving algorithms contains five chapters (7-11). In the first chapter the basic concept of algorithm (as point to set maps) are given and general convergence conditions are studied. The next chapter covers algorithms for unconstrained optimization. Here the classical topics up to quasi-newton methods are covered. Furthermore two sections are devoted to multidimensional search and another one to subgradient methods. The chapters nine consider the penalty and barrier methods. Here a special section is concerned with the polynomial-time convergence of interior point algorithms for linear programming. The next chapter deals with the methods of feasible directions, for instance SQP. A final chapter 11 covers problems with particular structures, as linear complementarity problems and quadratic, fractional or geometric programming. Algorithms exploiting the special characteristics of the models are discussed. A special section is devoted to the case of (including non convex) quadratic programming. In comparison with the second edition this third one contains some new topics (some of them mentioned above). More numerical examples and graphical illustrations are also included. Another new point is the more detailed treatment on modelling and the corresponding examples in the first chapter. In general the book remains a solid reference and a good textbook on the topic.

Reviewer: Walter Gómez Bofill (Temuco)

##### MSC:

90C30 | Nonlinear programming |

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

49J52 | Nonsmooth analysis |

90C20 | Quadratic programming |

90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |

90C32 | Fractional programming |

90C25 | Convex programming |

90C26 | Nonconvex programming, global optimization |

90C46 | Optimality conditions and duality in mathematical programming |

90C53 | Methods of quasi-Newton type |

90C55 | Methods of successive quadratic programming type |

49N15 | Duality theory (optimization) |