Convex and nonsmooth problems of vector optimization. (Vypuklye i negladkie zadachi vektornoj optimizatsii.) (Russian) Zbl 0765.90079

Gorokhovik, V. V.

Minsk: Navuka i Tekhnika. 240 p. (1990).

Two main topics of vector optimization are treated in this monograph: scalarization of convex problems and optimality conditions for nonsmooth problems. The material is arranged in an introduction and 5 chapters: 1) Sublinear preorder relations; 2) Scalarization of convex vector optimization problems; 3) Approximate quasidifferentiability of real- valued functions and optimality conditions in scalar optimization; 4) Approximate quasidifferentiability of vector-valued functions and optimality conditions in vector optimization; 5) Necessary optimality conditions for control problems with terminal vector objectives.
In the first chapter, preliminary concepts of binary relations are introduced; sublinear preorder relations are analyzed together with dual objects, the so-called corteges of linear functionals. In Chapter 2, two scalarization approaches are developed for convex vector problems: the first one is standard which uses linear functionals; the second one is nonconventional, which uses corteges (families of linear functionals) instead. Chapter 3 is devoted to the presentation of quasidifferential and \(\varepsilon\)-quasidifferential of real-valued functions and their applications in deriving optimality conditions for nonsmooth scalar problems. All this is then extended to the case of vector-valued functions in Chapter 4 in order to establish optimality conditions for vector problems. In the last chapter, optimal control problems with terminal vector objectives are tackled by using again quasidifferentials and \(\varepsilon\)-quasidifferentials.
The book contains the author’s deep investigations on optimality conditions in vector optimization via quasidifferentials. It is a valuable source for all researchers in the field of vector and nonsmooth optimization.

Reviewer: Dinh The Luc (Limoges)

MSC:

90C29	Multi-objective and goal programming
49J52	Nonsmooth analysis
90-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming

Keywords:

optimal control; sublinear preorder relations; approximate quasidifferentiability; vector optimization; scalarization of convex problems; optimality conditions for nonsmooth problems; optimality conditions; terminal vector objectives; nonsmooth optimization