Generalized convexity and vector optimization
Monograph devoted to various types of generalized convexity and their application to optimization and multi-objective programming problems. The text of the book is divided into 8 chapters. Chapter 1 contains the formulation of nonlinear programming problem and introduces the basic notions necessary in further parts of the book. Chapters 2 and 3 describe various approaches to generalization the notions of convex sets and functions. In this context invex sets, invex functions, quasi- and pseudoinvex functions, \(V\)-invex and univex functions as well as their modifications are defined. Both differentiable and non-differentiable generalizations are considered. Optimality conditions for multi-objective problems, fractional programming problems (both one-objective and multiobjective) and mathematical programming problems; under various types of generalized convexity are derived in Chapter 4. Chapter 5 deals with various approaches to duality theory for problems considered in Chapter 4. Especially Mond-Weir type duality and its generalization are considered. Chapter 6 investigates the second and higher-order duality results for minimax problems and some other non-differentiable optimization problems under generalized convexity assumptions. In Chapter 7 symmetric, and mixed-symmetric duality results for problems from previous chapters are presented. Especially mathematical programming problems, minimax mixed integer problems, and multi-objective non-differentiable fractional variational problems are investigated in this context. The last Chapter 8 investigates relationships between vector variational-like inequality problems and vector optimization problems under various generalizations of convexity whereas both smooth and non-smooth vector optimization problems are considered. The authors mention also motivations to generalizations of convexity. The motivation stemmed from the fact that the ``classical'' convexity assumptions in many cases do not satisfy the real-world requirements.
scientific article; zbMATH DE number 5358937