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Reviewer: Ionel Michael Navon

This is a very comprehensive monograph that uses Lagrange multiplier theory to present optimization, including constrained and convex optimization in functional spaces. It analyzes optimization methods involving partial differential equation (PDE) constraints. The treatment of variational problems is done in functional spaces, representing the view that omitting to do so and proceeding directly to discretization approaches results in unnecessary ill conditioning and loss of other structural properties. The book is written by experienced researchers in optimization and control theory, and PDE-constrained optimization. Chapter 1 provides the tools that establish the existence of Lagrange multipliers, with illustrative examples of optimal control problems involving various levels of difficulty. Chapter 2 presents important theoretical material related to sensitivity analysis of constrained nonlinear programming, along with Lipschitz continuity and differentiability of solutions of both optimization and optimal control problems with respect to optimization parameters. The theory is illustrated with an application to optimal control of an ordinary differential equation (ODE). Chapters 3, 5, and 6 constitute a unified logical unit that deals with smooth optimization issues. Chapter 3 deals in particular with augmented Lagrangian optimization for problems with both equality and inequality constraints. An illustrative application to a parameter estimation problem is provided. Chapters 5 and 6 concentrate on second-order iterative solution methods, such as Newton and sequential quadratic programming (SQP) for equality constraint problems. The results are then specialized for a class of optimal control problems that deal with the Navier-Stokes equations. Chapter 6 focuses on second-order augmented Lagrangian-SQP problems, where both state and control variables are treated as independent variables, while the optimality system is comprised of both primal and adjoint variables that are just the Lagrange multipliers (or adjoint variables) originating in the PDE constraints. This allows one to obtain the SQP method as a direct consequence. In order to convexify the optimization problem, a penalty term is added to the constraining equation. Section 6.4 provides examples of applications couched in functional space notation and specialized to optimal control problems governed by PDEs of the nonlinear elliptic type. Ito and Kunisch provide a mesh-independence result, asserting that under certain conditions, for SQP and Newton methods, the number of iterations required to attain a specified approximation criterion is independent of the mesh size. Useful comments and references to relevant literature are provided at the end of this chapter. Another large logical unit consists of chapters 4, 7, 8, and 9. These chapters deal with augmented Lagrangian methods for nonsmooth optimization and related concepts, the primal-dual active set method, semismooth Newton (SSN) methods, and their applications. Chapter 4 summarizes concepts of convex analysis for use in the subsequent chapters. In particular, the concept of subdifferential and the Fenchel duality theorem, along with the Yosida-Moreau approximation, are used as a basis for the primal-dual active set strategy, along with the augmented Lagrangian method. Various applications, such as image restoration, are used to illustrate the theory presented. Chapter 7 is dedicated to the primal-dual active set strategy for variational problems with simple constraints. It describes the methodology of both unilateral and bilaterally constrained problems. Chapter 8 deals with SSN methods; it first addresses the finite dimensional case, a Gauss-Newton algorithm, then proceeds to semismooth functions in the infinite dimensional spaces, and concludes with SSN methods with regularization. Applications using chapter 8's theory of SSN methods to nondifferentiable variational problems in function spaces are presented in chapter 9. In chapter 9, bounded variation image restoration problems are presented in a theoretical setup, while numerical experience with algorithmic concepts are referenced. The chapter then proceeds with the application of SSN methods to contact problems with Tresca and Coulomb friction in two spatial dimensions. Example 9.11 presents an illustrative algorithmic implementation for solving contact problems, with both Tresca and Coulomb friction, using both SSN and conceptual augmented Lagrangian ALM-FP algorithms, whose performance is investigated, along with the dependence of the friction coefficient. Chapter 10 is dedicated to the Lagrangian multiplier treatment of parabolic variational inequalities in unbounded domains. This is related in detail to the Black-Scholes model for American options. Chapter 11 focuses on the important problem of shape optimization. It provides a method for obtaining the shape derivative of the cost functional in shape optimization that obviates the need to use the shape derivative of the state variables of the system of PDEs. Also presented are the value problem, inverse interface problem, and stationary Navier-Stokes equations system. The book is very valuable for expert multidisciplinary researchers specialized in optimization theory, nonsmooth optimization, and PDE-constrained optimization or optimal control of PDEs. It contains a unique synthesis between optimization theory, optimal control problems, and inverse and parameter estimation problems, treated in the correct functional space setting. The examples provided are very useful, although the choice of discretization and final algorithmic implementation are left to the readers to find in the cited references. The book, written by two of the leading experts in the field, contains a wealth of up-to-date theoretical results that would otherwise require searching in many recent archival publications. As such, it will be a most welcome addition to many researchers' shelves. I strongly recommend it as a valuable addition to the array of tools available to researchers in the optimization and control theory field. Online Computing Reviews Service