Convex analysis and global optimization.

*(English)*Zbl 0904.90156
Nonconvex Optimization and Its Applications. 22. Dordrecht: Kluwer Academic Publishers. xi, 339 p. (1998).

The book is divided into two parts. The first part is a presentation of convex analysis, including the “traditional” analysis such as separation theorem, polar sets, and the study of convex functions, including approximate subdifferentials and duality. However this part on convex analysis also includes a chapter on DC (difference of convex) functions and sets, a class to which it is shown that many global optimization problems can be reduced.

Part II is entitled global optimization. The first chapter presents global optimality criteria, as well as DC inclusions associated with a feasible solution. The remaining chapters discuss various algorithmic procedures for finding global optima: partitioning methods (including concavity cuts), outer and inner approximations, decomposition, and the final chapter is devoted to nonconvex quadratic programming.

The book is carefully written and is a very useful introduction to the subject of global optimization. Since all basic concepts are presented in chapters on convex analysis, the prerequisites are very limited. Also exercises are stated the end of all chapters. Therefore the book may be used as a textbook for graduate courses. From the point of view of the practitioner we may regret for the absence of an overview of the impact of global optimization for real world optimization problems. Clearly this is the counterpart of the strong mathematical orientation of the book. Nevertheless, the book will be useful to practitioners that wish to have a good understanding of the basic principle of global optimization algorithm.

Part II is entitled global optimization. The first chapter presents global optimality criteria, as well as DC inclusions associated with a feasible solution. The remaining chapters discuss various algorithmic procedures for finding global optima: partitioning methods (including concavity cuts), outer and inner approximations, decomposition, and the final chapter is devoted to nonconvex quadratic programming.

The book is carefully written and is a very useful introduction to the subject of global optimization. Since all basic concepts are presented in chapters on convex analysis, the prerequisites are very limited. Also exercises are stated the end of all chapters. Therefore the book may be used as a textbook for graduate courses. From the point of view of the practitioner we may regret for the absence of an overview of the impact of global optimization for real world optimization problems. Clearly this is the counterpart of the strong mathematical orientation of the book. Nevertheless, the book will be useful to practitioners that wish to have a good understanding of the basic principle of global optimization algorithm.

Reviewer: J.F.Bonnans (Le Chesnay)

##### MSC:

90C30 | Nonlinear programming |

49J52 | Nonsmooth analysis |

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

90C26 | Nonconvex programming, global optimization |

65K05 | Numerical mathematical programming methods |