Convexity has been increasingly important in recent years in the studyof extremum problems in ,many areas of applied mathematics. The purposeof this book is to provide an exposition of the theory of convex sets andfunctions in which applications to extremum problems play the centralrole.
Preface
Introductory Remarks:a Guide for the Reader
PART Ⅰ:BASIC CONCEPTS
§1.Affine Sets
§2.Convex Sets and Cones
§3.The Algebra of Convex Sets
§4.Convex Functions
§5.Functional Operations
PART Ⅱ:TOPOLOGICAL PROPERTIES
§6.Relative Interiors of Convex Sets
§7.Closures of Convex Functions
§8.Recession Cones and Unboundedness
§9.Some Closedness Criteria
§10.Continuity of Convex Functions
PART Ⅲ:DUALITY CORRESPONDENCES
§11.Separation Theorems
§i2.Conjugates of Convex Functions
§13.Support Functions
§14.Polars of Convex Sets
§15.Polars of Convex Functions
§16.Dual Operations
PART Ⅳ:REPRESENTATION AND INEQUALITIES
§i7.Caratheodory's Theorem
§18.Extreme Points and Faces of Convex Sets
§19.Polyhedral Convex Sets and Functions
§20.Some Applications of Polyhedral Convexity
§21.Helly's Theorem and Systems of Inequalities
§22.Linear Inequalities
PART Ⅴ:DIFFERENTIAL THEORY
§23.Directional Derivatives and Subgradients
§24.Differential Continuity and Monotonicity
§25.Differentiability of Convex Functions
§26.The Legendre Transformation
PART Ⅵ:CONSTRAINED EXTREMUM PROBLEMS
§27.The Minimum of a Convex Function
§28.Ordinary Convex Programs and Lagrange Multipliers
§29.Bifunctions and Generalized Convex Programs
§30.Adjoint Bifunctions and Dual Programs
§31.Fenchers Duality Theorem
§32.The Maximum of a Convex Function
PART Ⅶ:SADDLE-FUNCTIONS AND MINIMAX THEORY
§33.Saddle-Functions
§34.Closures and Equivalence Classes
§35.Continuity and Differentiability of Saddle-functions
§36.Minimax Problems
§37.Conjugate Saddle-functions and Minimax Theorems
PART Ⅷ:CONVEX ALGEBRA
§38.The Algebra of Bifunctions
§39.Convex Processes
Comments and References
Bibliography
Index