Theory of vector optimization. (English) Zbl 0654.90082

Dinh The Luc

Lecture Notes in Economics and Mathematical Sciences, 319. Berlin etc.: Springer-Verlag. viii, 173 p. (1988).

Since the fifties vector optimization has become a mathematical discipline and today we witness an increasing development of this field with a large number of results and applications. In this book, written by a well-known author in the field mentioned, a systematic study of theoretical aspects of vector optimization with the emphasis on nonconvex problems in infinite-dimensional spaces ordered by convex cones is attempted.
The chapters are entitled as follows: (1) analysis over cones; (2) efficient points and vector optimization problems; (3) nonsmooth vector optimization problems; (4) scalarization and stability; (5) duality; (6) structure of optimal solutions sets. Historical and bibliographical comments, almost 220 references, an index and a preface complete this book.
The first chapter contains properties of cones with special structures, cone monotonic, cone convex and cone continuous functions as well as remarks on set-valued mappings. - The second chapter deals with partial orders and with the existence of efficient points. Remarks on external stability of the set of efficient points are included too. - In the third chapter contingent derivatives are chosen to produce optimality conditions for vector problems with set-valued data. In the last sections of this chapter the general results obtained are applied to the special case of point-valued Fréchet differentiable functions. - The fourth chapter begins with separation by monotonic functions followed by a number of scalarization results and, using set-valued maps, stability results. - The fifth chapter is devoted to duality theory of vector problems with set-valued objectives in a very general setting. Classical approaches such as Lagrangian and conjugacy concepts as well as axiomatic approaches are dealt with. The last section is of special interest: The equivalence between duality and alternative properties is established and applied to examples working in lattices. - In the sixth chapter conditions for closedness, connectedness or contractibility of efficient point sets are given.
The book is written very clearly with full proofs, comments and examples, and so I think that it might be a good background for applications and further developments in vector optimization.

Reviewer: A.Göpfert

MSC:

90C31	Sensitivity, stability, parametric optimization
90-02	Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming
49N15	Duality theory (optimization)

Keywords:

vector optimization; nonconvex problems in infinite-dimensional spaces; convex cones; set-valued mappings; existence of efficient points; contingent derivatives; scalarization; stability; duality theory of vector problems