Smoothing is a technique used in solving many optimization problems. This book introduces semismooth Newton methods, a type of smoothing, and discusses related analysis, including basic equations, derived equations, algorithms, theorems, proofs of the theorems, and numerical results. The book applies these methods to various optimization problems.

Chapter 1 introduces the concept of semismooth Newton methods for finite- and infinite-dimensional variational inequalities. It presents a list of properties and examples of the semismooth Newton methods, including an optimal control problem and an obstacle problem; basic equations are set up for these problems. The chapter ends with an overview of the organization of the book. Chapter 2 formally defines semismoothness and the semismooth Newton method. The method is defined as an iterative algorithm to solve an equation. For faster convergence of the algorithm, higher-order semismoothness is introduced. The chapter also presents examples of semismooth functions, such as piecewise differentiable functions. Chapter 3 presents various algorithms of semismooth Newton methods for various operator equations, including semismooth operators in Banach spaces, semismooth superposition operators, semismooth composite operators, and generalized differentials. The analysis and associated proofs are also discussed. Chapter 4 explains smoothing steps and describes the derivation of a formula for a smoothing step. It also presents a semismooth Newton method that does not have any smoothing steps. The chapter ends by deriving a related sufficiency condition.

Chapter 5 applies the semismooth techniques developed in earlier chapters to more general variational inequality problems, including bound-constrained variational inequality problems, pointwise convex constrained variational inequality problems, and Karush-Kuhn-Tucker (KKT) systems. The discussion of KKT systems includes the way the Newton method is applied, the related smoothing steps, and related regularities. Chapter 6 is on mesh independence, which refers to algorithm convergence. Mesh independence is explained under various types of conditions, such as uniform growth conditions, without uniform growth conditions, and without growth conditions. Some theoretical results are presented for both general semismoothness and Newton semismoothness. The results are applied to control-constrained semi-linear elliptic optimal control problems. Chapter 7 describes the trust region algorithm, which is global in nature, and associated algorithms for trust region radius update and reduction ratio computation. The chapter proves the related global convergence and ends by describing a trust-region projected Newton algorithm for fast local convergence.

Chapter 8 discusses a class of convex optimization problems with state constraints, such as optimal control problems and elliptical obstacle problems. Regularization involves using penalty techniques, and the author proves convergence and rate of convergence for these penalty techniques. Chapter 9 applies Newton semismoothness to the distributed control of a semi-linear elliptic equation problem, using approaches such as block box and all at once. Numerical results for these approaches are also discussed, and the author applies the semismooth Newton methods to obstacle problems.

The last two chapters are on Navier-Stokes fluid flow problems. Chapter 10, on incompressible flows, sets up the equations for functional analytics. The problem is reduced using control state mappings, so that Newton semismoothness can be applied. The chapter ends with a discussion of numerical results for the pointwise bound constraints and the pointwise ball constraints. Chapter 11 describes an algorithm to solve compressible flow. The solution uses adjoint-based gradient computation. Numerical results for the algorithm are also discussed. An appendix describes the adjoint approach in detail.

The book is written for readers with a strong background in optimization problems, differential and integral calculus, algorithms, numerical computations, discrete structures, matrices, and statistics. It could also be used as a text in a graduate-level course. Professionals solving problems requiring optimization will also benefit from the book, as will researchers with theoretical and computational interests in related problems.