Consisting of two parts, this work presents the main notions of abstract convexity and their applications in the study of some classes of functions and sets. It examines both theoretical and numerical aspects of global optimization based on abstract convexity. It is of interest to specialists in global optimization, and mathematical programming.

This book consists of two parts. Firstly, the main notions of abstract convexity and their applications in the study of some classes of functions and sets are presented. Secondly, both theoretical and numerical aspects of global optimization based on abstract convexity are examined. Most of the book does not require knowledge of advanced mathematics. Classical methods of nonconvex mathematical programming, being based on a local approximation, cannot be used to examine and solve many problems of global optimization, and so there is a clear need to develop special global tools for solving these problems.Some of these tools are based on abstract convexity, that is, on the representation of a function of a rather complicated nature as the upper envelope of a set of fairly simple functions. This book will be of interest to specialists in global optimization, mathematical programming, and convex analysis, as well as engineers using mathematical tools and optimization techniques and specialists in mathematical modelling.

Preface. Acknowledgment. 1. An Introduction to Abstract Convexity. 2. Elements of Monotonic Analysis: IPH Functions and Normal Sets. 3. Elements of Monotonic Analysis: Monotonic Functions. 4. Application to Global Optimization: Lagrange and Penalty Functions. 5. Elements of Star-Shaped Analysis. 6. Supremal Generators and Their Applications. 7. Further Abstract Convexity. 8. Application to Global Optimization: Duality. 9. Application to Global Optimization: Numerical Methods. References. Index.