Convexity and optimization in Banach spaces.
4th updated and revised ed.

*(English)*Zbl 1244.49001
Springer Monographs in Mathematics. Dordrecht: Springer (ISBN 978-94-007-2246-0/hbk; 978-94-007-2247-7/ebook). xii, 368 p. (2012).

The authors present the fourth English edition of their book “Convexity and optimization in Banach spaces”.
[For reviews of earlier editions see e.g., Romanian original (1975; Zbl 0317.49011), Revised and enlarged translation into
English (1978; Zbl 0379.49010), and 2nd revised and extended translation into English (1986; Zbl 0594.49001).]

This editon contains new results concerning subdifferential calculus for convex functions and for duality in convex programming. The last chapter was rewritten for this edition. The book presents many of the fundamental results of the theory of infinite-dimensional convex analysis which were obtained in the last 25 years.

The book provides the reader with useful tools concerning convex analysis. The content of the book is organized in four chapters.

Chapter 1 (Fundamentals of functional analysis): convexity in topological linear spaces, duality in linear normed spaces, vector-valued functions and distributions, maximal monotone operators and evolution systems in Banach spaces. Chapter 2 (Convex functions): general properties of convex functions, the subdifferential of a convex function, concave-convex functions. Chapter 3 (Convex programming): optimality conditions, duality in convex programming, application of the duality theory. Chapter 4 (Convex control problems in Banach spaces): distributed optimal control problems, synthesis of optimal control, boundary optimal control problems, optimal control problems on a half-axis, optimal control of linear periodic resonant systems.

Every chapter ends with problems, bibliographical notes and references.

This editon contains new results concerning subdifferential calculus for convex functions and for duality in convex programming. The last chapter was rewritten for this edition. The book presents many of the fundamental results of the theory of infinite-dimensional convex analysis which were obtained in the last 25 years.

The book provides the reader with useful tools concerning convex analysis. The content of the book is organized in four chapters.

Chapter 1 (Fundamentals of functional analysis): convexity in topological linear spaces, duality in linear normed spaces, vector-valued functions and distributions, maximal monotone operators and evolution systems in Banach spaces. Chapter 2 (Convex functions): general properties of convex functions, the subdifferential of a convex function, concave-convex functions. Chapter 3 (Convex programming): optimality conditions, duality in convex programming, application of the duality theory. Chapter 4 (Convex control problems in Banach spaces): distributed optimal control problems, synthesis of optimal control, boundary optimal control problems, optimal control problems on a half-axis, optimal control of linear periodic resonant systems.

Every chapter ends with problems, bibliographical notes and references.

Reviewer: Erich Miersemann (Leipzig)

##### MSC:

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

49J27 | Existence theories for problems in abstract spaces |

49K27 | Optimality conditions for problems in abstract spaces |

46A55 | Convex sets in topological linear spaces; Choquet theory |

49J40 | Variational inequalities |

49J45 | Methods involving semicontinuity and convergence; relaxation |

49N15 | Duality theory (optimization) |

49K35 | Optimality conditions for minimax problems |

90C25 | Convex programming |

47H05 | Monotone operators and generalizations |

49J35 | Existence of solutions for minimax problems |