Set-valued Optimization: An Introduction with Applications (Vector Optimization)

Set-valued optimization is a vibrant and expanding branch of mathematics that deals with optimization problems where the objective map and/or the constraints maps are set-valued maps acting between certain spaces. Since set-valued maps subsumes single valued maps, set-valued optimization provides an important extension and unification of the scalar as well as the vector optimization problems. Therefore this relatively new discipline has justifiably attracted a great deal of attention in recent years. This book presents, in a unified framework, basic properties on ordering relations, solution concepts for set-valued optimization problems, a detailed description of convex set-valued maps, most recent developments in separation theorems, scalarization techniques, variational principles, tangent cones of first and higher order, sub-differential of set-valued maps, generalized derivatives of set-valued maps, sensitivity analysis, optimality conditions, duality and applications in economics among other things. Front Matter....Pages i-xxii Introduction....Pages 1-10 Order Relations and Ordering Cones....Pages 11-76 Continuity and Differentiability....Pages 77-108 Tangent Cones and Tangent Sets....Pages 109-211 Nonconvex Separation Theorems....Pages 213-248 Hahn-Banach Type Theorems....Pages 249-273 Conjugates and Subdifferentials....Pages 275-306 Duality....Pages 307-348 Existence Results for Minimal Points....Pages 349-368 Ekeland Variational Principle....Pages 369-397 Derivatives and Epiderivatives of Set-Valued Maps....Pages 399-508 Optimality Conditions in Set-Valued Optimization....Pages 509-604 Sensitivity Analysis in Set-Valued Optimization and Vector Variational Inequalities....Pages 605-643 Numerical Methods for Solving Set-Valued Optimization Problems....Pages 645-661 Applications....Pages 663-725 Back Matter....Pages 727-765