Vector optimization. Theory, applications, and extensions. (English) Zbl 1055.90065

Jahn, Johannes

Berlin: Springer (ISBN 3-540-20615-9/hbk). xiii, 465 p. (2004).

J. Jahn, well known by his papers and books on convex analysis and optimization (mostly in general spaces), wrote this interesting book that gives a clear insight into theory and application of vector optimization. It is not only a revised version of his book from 1986 (see Zbl 0578.90048) but he also extended the contents considerably (very often supported by his own results) with respect to engineering applications (up to computed solutions, having used a modified Polak method, a weighted Chebyshev norm approach and the reference point approximation) and to the recently growing field of set optimization. Set optimization here means optimization with set-valued constraints or a set-valued objective function. It is shown, that this branch of optimization is not only very interesting but also closely related to problems in e.g. stochastic programming or optimal control problems with differential inclusions. The basic concepts of set optimization, especially the used partial orderings, contingent epiderivatives and subdifferentials, are clearly explained and followed by presentation of optimality conditions of Lagrange multiplier rule type.

Reviewer: Alfred Göpfert (Halle)

MSC:

90C29	Multi-objective and goal programming
90-02	Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
90C48	Programming in abstract spaces
49J53	Set-valued and variational analysis
90C90	Applications of mathematical programming
58E17	Multiobjective variational problems, Pareto optimality, applications to economics, etc.

Keywords:

vector optimization; set optimization; Lagrange multiplier rule; convex analysis; duality

Citations:

Zbl 0578.90048