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This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space. The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.
List of contents
1 Variable ordering structures.- 2 Optimality concepts and their characterization.- 3 Properties of cone-valued maps.- 4 Linear scalarizations.- 5 Nonlinear scalarizations.- 6 Scalarization for variable orderings given by Bishop-Phelps cones.- 7 Optimality conditions.- 8 Duality results.- 9 Numerical methods.- 10 Outlook and further application areas.
About the author
Gabriele Eichfelder is Professor for mathematical methods of Operations Research at Technical University of Ilmenau, Germany.
Summary
This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space. The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.
Additional text
“This book is the first comprehensive introduction
to the particular topic of vector optimization with variable ordering
structures. … The presentation of the book is very detailed and relatively easy
for reading. … Graduate students and researchers will find this book useful as
a standard reference on vector optimization with variable ordering structures.”
(Phan Quốc Khánh, Mathematical Reviews, December, 2015)
Report
"This book is the first comprehensive introduction to the particular topic of vector optimization with variable ordering structures. ... The presentation of the book is very detailed and relatively easy for reading. ... Graduate students and researchers will find this book useful as a standard reference on vector optimization with variable ordering structures." (Phan Qu c Khánh, Mathematical Reviews, December, 2015)
