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

*(Russian)*Zbl 0765.90079
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.

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 |