zbMATH — the first resource for mathematics

Lagrange multiplier approach to variational problems and applications. (English) Zbl 1156.49002
Advances in Design and Control 15. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-0-898716-49-8/pbk; 978-0-89871-861-4/ebook). xvi, 341 p. (2008).
In this interesting monographical book infinite-dimensional analysis is used to study in a comprehensive way Lagrange multiplier approach to nonlinear variational problems with equality and (with help of cones) inequality constraints. The contents of the book can be characterized by three intimately connected points:
Existence of Lagrange multipliers and sensitivity analysis (including convex analysis),
Computational methods and their behaviour for different classes of the considered variational problems,
Profound examples, which fit in the framework of both the first points.
Just these examples, their treatment together with solution techniques, make the book suspenseful and exciting. One finds optimal control problems, structural optimization, inverse, contact, obstacle, Signorini and friction problems, image reconstruction, mathematical finance, problems with constraining partial differential equations, problems with free boundaries, elliptic, parabolic and Navier-Stokes equations. The book is clearly written, with full proofs and it is convenient, that it starts with a rich in content preface (of seven pages).

49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
49J27 Existence theories for problems in abstract spaces
49K27 Optimality conditions for problems in abstract spaces
49K40 Sensitivity, stability, well-posedness
49M30 Other numerical methods in calculus of variations (MSC2010)
49M37 Numerical methods based on nonlinear programming
49N15 Duality theory (optimization)
90C48 Programming in abstract spaces
Full Text: DOI