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**Convex optimization.**
*(English)*
Zbl 1058.90049

Cambridge University Press (ISBN 0-521-83378-7/hbk). xiii, 716 p. (2004).

This book presents a comprehensive and rather complete study of convex optimisation in a single volume. Theory and procedures that conform a bridge to applications are provided. The theoretical presentations include rigorous proofs and clarifying discussions, which include the development of examples. A list of complementary exercises is given at the end of each chapter. The authors are leading figures in this field. A large list of references, around 170, is provided. It is divided into 3 parts, which are preceded by an introductory chapter.

Part 1 begins with the derivation of theoretical aspects of convexity. After studying the convexity of sets and functions as well as the realm of general convex optimization, duality is analysed at large.

The second part is concerned with applications. It is particularly interesting. Its first chapter, the sixth in the general index, deals with the problematic of approximating data trough a mathematical model. They consider that the task of optimisers is to determine a consistent model to describe prior information, looking for a minimising a measure of a certain approximation error. The second one presents how convex optimization is interrelated with the solution of different statistical estimation problems. The optimisation point of view of the correct treatment of Maximum Likelihood vector estimation may pose particularly complicated problems. The authors considered them and a certain convex optimisation model, which allows predicting or estimating, is presented and the properties are discussed. The particular statistical problems studied are of wide interest. They as general as the difficulties arising in the computation of predictions when a noise is present and in covariance estimation under Normal distributions; and as concrete as the discrete problems concerned with estimation under Poisson generated counting processes and logistic regression fitting. The estimation of a distribution function [a priori distribution for Bayesian analysis and for using in a non-parametric approach] represents another set of frequently complicated estimation problems that they study using convex optimisation. Optimal design of experiments, robust detection and bounding, constitute another considerably large set of models with which statistical theoreticians should deal rending to optimisation the task of deriving the adequate numerical solution. Convex optimisation models that can achieve solutions to them are concisely discussed in this chapter. Chapter 8 presents how convex optimization copes with geometrical problems. The survey includes the projection of a point in a convex set, the computation of intra set distances and separations of convex sets.

The main feature of Part 3 [algorithms] is the presentation of simple and reliable algorithms for solving unconstrained, equally constrained and inequality constrained convex optimization programs. They are based on Newton type methods and interior point methods.

Three appendixes ar included. They provide the needed mathematical and numerical linear algebra knowledge for coping with the theoretical contents together with a brief presentation of some convex optimisation problems that play an important role in applications.

Part 1 begins with the derivation of theoretical aspects of convexity. After studying the convexity of sets and functions as well as the realm of general convex optimization, duality is analysed at large.

The second part is concerned with applications. It is particularly interesting. Its first chapter, the sixth in the general index, deals with the problematic of approximating data trough a mathematical model. They consider that the task of optimisers is to determine a consistent model to describe prior information, looking for a minimising a measure of a certain approximation error. The second one presents how convex optimization is interrelated with the solution of different statistical estimation problems. The optimisation point of view of the correct treatment of Maximum Likelihood vector estimation may pose particularly complicated problems. The authors considered them and a certain convex optimisation model, which allows predicting or estimating, is presented and the properties are discussed. The particular statistical problems studied are of wide interest. They as general as the difficulties arising in the computation of predictions when a noise is present and in covariance estimation under Normal distributions; and as concrete as the discrete problems concerned with estimation under Poisson generated counting processes and logistic regression fitting. The estimation of a distribution function [a priori distribution for Bayesian analysis and for using in a non-parametric approach] represents another set of frequently complicated estimation problems that they study using convex optimisation. Optimal design of experiments, robust detection and bounding, constitute another considerably large set of models with which statistical theoreticians should deal rending to optimisation the task of deriving the adequate numerical solution. Convex optimisation models that can achieve solutions to them are concisely discussed in this chapter. Chapter 8 presents how convex optimization copes with geometrical problems. The survey includes the projection of a point in a convex set, the computation of intra set distances and separations of convex sets.

The main feature of Part 3 [algorithms] is the presentation of simple and reliable algorithms for solving unconstrained, equally constrained and inequality constrained convex optimization programs. They are based on Newton type methods and interior point methods.

Three appendixes ar included. They provide the needed mathematical and numerical linear algebra knowledge for coping with the theoretical contents together with a brief presentation of some convex optimisation problems that play an important role in applications.

Reviewer: Carlos Narciso Bouza Herrera (Habana)

### MSC:

90C25 | Convex programming |

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

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

90C90 | Applications of mathematical programming |