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Reviewer: Florin Popentiu

Presenting the current methods used in nonlinear optimization, this book is a comprehensive presentation of different fields of optimization theory: linear, integer, stochastic, non-smooth optimization, optimal control, and semi-infinite programming. The specific subjects are oriented toward the following directions of research: optimality conditions, duality theory, sensitivity analysis, and numerical methods. According to the author: Optimization problems arise in approximation theory, probability theory, structure design, chemical process control, routing in telecommunication networks, image reconstruction, experiment design, radiation therapy, asset valuation, portfolio management, supply chain management, [and] facility location, among others. Organized into two parts and an appendix, the seven chapters provide recent information in this very successful area of applied mathematics. Following the most important theoretical and methodological acquisitions, the book seeks to foster an understanding of the modern ideas and principles of optimization, in a coherent way. Part 1 discusses the theory of optimization, including an "Introduction," "Elements of Convex Analysis," "Optimality Conditions," and "Lagrangian Duality." After the introduction in chapter 1, chapter 2 presents the theoretical foundations of convex functions, highlighting the elements that are necessary for optimization theory. The optimality conditions are used in the third chapter to analyze the conditions that have to be satisfied at points that are local minima. The fourth chapter introduces the duality theory (Lagrangian duality) for nonlinear optimization problems. Part 2 (chapters 5 to 7) makes use of the theory presented in the first part in order to frame numerical methods of optimization, and to analyze their relevance and efficiency. Chapter 5 approaches the unconstrained optimization of differentiable functions; after an introduction to iterative algorithms, important basic methods are detailed: Newton's method, conjugate gradient, quasi-Newton, trust region, and nongradient methods. Chapter 6 is dedicated to constrained optimization of differentiable functions, and presents popular techniques to solve these kinds of problems. The author approaches feasible point, penalty, basic dual, and augmented Lagrangian methods, continuing with Newton's method and barrier methods, with detailed and easy-to-follow proofs that are illustrated by numerous examples and figures. In chapter 7, the author approaches nondifferentiable optimization, illustrated by subgradient, cutting plane, proximal point, bundle, and trust-region methods. There is also a discussion of aspects related to constrained problems, composite optimization, and nonconvex constraints; the complexity of the problems is detailed. Finally, an outline of the stability of set-constrained systems is given in Appendix A. Owing to its wide range of nonlinear optimization topics, this book will be useful to all specialists interested in modern topics such as optimality conditions and numerical methods for problems involving nondifferentiable functions, semidefinite programming, metric regularity and stability theory of set-constrained systems, and sensitivity analysis of optimization problems. Graduate students and researchers can use the research material as a starting point in their investigations. The references are very good. In my opinion, this outstanding work has great value. It will be useful to people working and teaching in the field of operational research, as it covers important information for researchers and practitioners. Online Computing Reviews Service