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**Numerical optimization.
2nd ed.**
*(English)*
Zbl 1104.65059

Springer Series in Operations Research and Financial Engineering. New York, NY: Springer (ISBN 0-387-30303-0/hbk). xxii, 664 p. (2006).

This review is devoted to the second edition of this book [Numerical optimization. Springer Series in Operations Research. NY: Springer (1999; Zbl 0930.65067)]; the review of the first edition remaining still as a reference point. This revised and updated edition of the book establishes new ‘weights’ for the enclosed fields and topics, adds new material in many places, and reflects a better understanding of constrained optimization viewed from the perspective of theoretical, algorithmic, and computational approaches. Two new chapters are enclosed in this edition: Chapter 9 deals with derivation-free optimization, which is of great interest in applications, and Chapter 19 is devoted to interior-point methods for nonlinear programming, a basic domain in successful codes for nonlinear programming.

It should be instructive to list the titles of all the comprised chapters: Chapter 1: Introduction, Chapter 2: Fundamentals of unconstrained optimization; Chapter 3: Line search methods; Chapter 4: Trust-region methods; Chapter 5: Conjugate gradient methods; Chapter 6: Quasi-Newton methods; Chapter 7: Large-scale unconstrained optimization; Chapter 8: Calculating derivatives; Chapter 9 (new): Derivative-free optimization; Chapter 10: Least-squares problems; Chapter 11: Nonlinear equations; Chapter 12: Theory of constrained optimization; Chapter 13: Linear programming: The simplex methods; Chapter 14: Linear programming: Interior-point methods; Chapter 15: Fundamentals of algorithms for nonlinear constrained optimization; Chapter 16: Quadratic programming; Chapter 17: Penalty and augmented Lagrangian methods; Chapter 18: Sequential quadratic programming; Chapter 19 (new): Interior-point methods for nonlinear programming; Appendices: A1. Elements of linear algebra; A2. Elements of analysis, geometry, topology; B. A Regularization procedure.

Apart from the new introduced chapters 9 and 19, a lot of topics received a special attention or have been reorganized in slightly different frameworks. E. g., (a) the discussion on Newton’s unconstrained optimization methods is distributed more naturally within several chapters, with a special emphasis on large-scale problems in Chapter 7. (b) Sequential quadratic programming and interior-point methods appear in Chapter 15 of nonlinear programming algorithms, while primal barrier methods moved to the new Chapter 19 on interior-point techniques for nonlinear programming. New material is added: (c) a treatment of nonlinear programming duality; (d) an expended discussion of algorithms for inequality constrained quadratic programming; (e) an approach of dual simplex and presolving in linear programming; (f) a summary of practical issues in the implementation of interior-point linear programming algorithms; (g) a description of conjugate-gradient methods for quadratic programming; (h) a discussion of filter methods and non-smooth penalty methods in nonlinear programming algorithms etc. The exercises attached at the end of each chapter have been revised in most chapters, and a manual containing solutions for selected problems will be available to bona fide instructors through the publisher. Moreover, to many chapters have been added an useful Perspectives and Software section, for an update discussion on the state-of-the-art in application software. As mentioned in the review to the first edition the users of optimization software may find at the link

http://www.mcs.anl.gov/otc/Guide/SoftwareGuide/

all the new packages and latest changes to existing software.

It should be instructive to list the titles of all the comprised chapters: Chapter 1: Introduction, Chapter 2: Fundamentals of unconstrained optimization; Chapter 3: Line search methods; Chapter 4: Trust-region methods; Chapter 5: Conjugate gradient methods; Chapter 6: Quasi-Newton methods; Chapter 7: Large-scale unconstrained optimization; Chapter 8: Calculating derivatives; Chapter 9 (new): Derivative-free optimization; Chapter 10: Least-squares problems; Chapter 11: Nonlinear equations; Chapter 12: Theory of constrained optimization; Chapter 13: Linear programming: The simplex methods; Chapter 14: Linear programming: Interior-point methods; Chapter 15: Fundamentals of algorithms for nonlinear constrained optimization; Chapter 16: Quadratic programming; Chapter 17: Penalty and augmented Lagrangian methods; Chapter 18: Sequential quadratic programming; Chapter 19 (new): Interior-point methods for nonlinear programming; Appendices: A1. Elements of linear algebra; A2. Elements of analysis, geometry, topology; B. A Regularization procedure.

Apart from the new introduced chapters 9 and 19, a lot of topics received a special attention or have been reorganized in slightly different frameworks. E. g., (a) the discussion on Newton’s unconstrained optimization methods is distributed more naturally within several chapters, with a special emphasis on large-scale problems in Chapter 7. (b) Sequential quadratic programming and interior-point methods appear in Chapter 15 of nonlinear programming algorithms, while primal barrier methods moved to the new Chapter 19 on interior-point techniques for nonlinear programming. New material is added: (c) a treatment of nonlinear programming duality; (d) an expended discussion of algorithms for inequality constrained quadratic programming; (e) an approach of dual simplex and presolving in linear programming; (f) a summary of practical issues in the implementation of interior-point linear programming algorithms; (g) a description of conjugate-gradient methods for quadratic programming; (h) a discussion of filter methods and non-smooth penalty methods in nonlinear programming algorithms etc. The exercises attached at the end of each chapter have been revised in most chapters, and a manual containing solutions for selected problems will be available to bona fide instructors through the publisher. Moreover, to many chapters have been added an useful Perspectives and Software section, for an update discussion on the state-of-the-art in application software. As mentioned in the review to the first edition the users of optimization software may find at the link

http://www.mcs.anl.gov/otc/Guide/SoftwareGuide/

all the new packages and latest changes to existing software.

Reviewer: Neculai Curteanu (Iaşi)

### MSC:

65K05 | Numerical mathematical programming methods |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

90C06 | Large-scale problems in mathematical programming |

90C30 | Nonlinear programming |

90C55 | Methods of successive quadratic programming type |

90C05 | Linear programming |

90C20 | Quadratic programming |

90C51 | Interior-point methods |

90C56 | Derivative-free methods and methods using generalized derivatives |