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**Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces.**
*(English)*
Zbl 1235.49001

Semismooth Newton methods occupy a central role in mathematics. The author has presented a remarkable selfcontained treatment of a large class of methods for the solution of optimization problems with PDE and inequality constraints combined with variational inequalities in function spaces. The book manuscript has originated from the author’s habilitation thesis. It provides a very successful attempt to this very active domain of contemporary research.

The book consists of 11 Chapters. Chapter 1, Introduction, is devoted to the presentation of basic theory for the development and analysis of a class of Newton-type methods for the solution of optimization problems as well as variational inequalities in the framework of function spaces and pointwise inequality constraints. Chapter 2 provides a discussion of essential results of nonsmooth analysis for mappings in finite dimensional spaces. In particular, a number of generalized differentials, including Clarke’s generalized Jacobian, B-differential as well as Qi’s C-subdifferential, which are studied in various contexts in the literature, are presented here. Newton methods as well as very carefully selected examples are nicely presented. In addition, semismoothness properties are studied for operator equations in Banach spaces as well as semismoothness of higher order. Several variants of these results have also been considered.

Chapter 3 is concerned with an application of the abstract semismoothness concept to operators that are obtained by superposition of a Lipschitz continuous semismooth function and a smooth operator mapping into a product of Lebesgue spaces. Chapter 4 studies a particular choice of the MCP-function that leads to reformulations for which no smoothing step is required. Chapter 5 discusses techniques for the solution of some more general types of problems than NCP’s. Emphasis is given to mixed type problems. Chapter 6 gives an account of techniques for the study of mesh-independence results of semismooth Newton methods for complementarity methods in \(L_p\) spaces. This theory for the classical Newton method cannot be directly extended to the semismooth case. For this reason some new techniques are discussed to develop mesh-independence results for semismooth Newton methods.

Chapter 7 deals with a discussion of an approach to make the developed class of semismooth Newton methods globally convergent by embedding them in a trust-region method. Chapter 8 studies some state-constrained optimal control and related problems. Furthermore an approach is introduced for regularizing the problem such that smooth or semismooth reformulations of the optimality system are possible. Chapter 9 is devoted to an application of developed algorithms to concrete problems. In particular, the applicability of semismooth Newton methods to a semilinear elliptic control problem with bounds on the control is studied. Chapter 10 contains a proof of the fact that the class of semismooth Newton methods can be applied for the solution of control-constrained distributed optimal control problems governed by incompressible Navier-Stokes equations. Chapter 11 deals with applications of the methods previously discussed to the boundary control of time-dependent compressible Navier–Stokes equations. In addition, the author has provided numerical results to prove that the approach used is viable and efficient also for quite large scale, state-of-the-art control problems.

The book concludes with some very interesting and useful supplements. The author has presented a self-contained survey of semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces. It provides a successful attempt to present a large amount of old and new research in a friendly way so that it proves to be useful not only for mathematical experts but for graduate students as well.

I strongly recommend the book for graduate seminars as well as a reference book.

The book consists of 11 Chapters. Chapter 1, Introduction, is devoted to the presentation of basic theory for the development and analysis of a class of Newton-type methods for the solution of optimization problems as well as variational inequalities in the framework of function spaces and pointwise inequality constraints. Chapter 2 provides a discussion of essential results of nonsmooth analysis for mappings in finite dimensional spaces. In particular, a number of generalized differentials, including Clarke’s generalized Jacobian, B-differential as well as Qi’s C-subdifferential, which are studied in various contexts in the literature, are presented here. Newton methods as well as very carefully selected examples are nicely presented. In addition, semismoothness properties are studied for operator equations in Banach spaces as well as semismoothness of higher order. Several variants of these results have also been considered.

Chapter 3 is concerned with an application of the abstract semismoothness concept to operators that are obtained by superposition of a Lipschitz continuous semismooth function and a smooth operator mapping into a product of Lebesgue spaces. Chapter 4 studies a particular choice of the MCP-function that leads to reformulations for which no smoothing step is required. Chapter 5 discusses techniques for the solution of some more general types of problems than NCP’s. Emphasis is given to mixed type problems. Chapter 6 gives an account of techniques for the study of mesh-independence results of semismooth Newton methods for complementarity methods in \(L_p\) spaces. This theory for the classical Newton method cannot be directly extended to the semismooth case. For this reason some new techniques are discussed to develop mesh-independence results for semismooth Newton methods.

Chapter 7 deals with a discussion of an approach to make the developed class of semismooth Newton methods globally convergent by embedding them in a trust-region method. Chapter 8 studies some state-constrained optimal control and related problems. Furthermore an approach is introduced for regularizing the problem such that smooth or semismooth reformulations of the optimality system are possible. Chapter 9 is devoted to an application of developed algorithms to concrete problems. In particular, the applicability of semismooth Newton methods to a semilinear elliptic control problem with bounds on the control is studied. Chapter 10 contains a proof of the fact that the class of semismooth Newton methods can be applied for the solution of control-constrained distributed optimal control problems governed by incompressible Navier-Stokes equations. Chapter 11 deals with applications of the methods previously discussed to the boundary control of time-dependent compressible Navier–Stokes equations. In addition, the author has provided numerical results to prove that the approach used is viable and efficient also for quite large scale, state-of-the-art control problems.

The book concludes with some very interesting and useful supplements. The author has presented a self-contained survey of semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces. It provides a successful attempt to present a large amount of old and new research in a friendly way so that it proves to be useful not only for mathematical experts but for graduate students as well.

I strongly recommend the book for graduate seminars as well as a reference book.

Reviewer: Basilis Kokkinis (Athens)

### MSC:

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

49M15 | Newton-type methods |

49J40 | Variational inequalities |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65K15 | Numerical methods for variational inequalities and related problems |