Optimality Conditions in Convex Optimization: A Finite-Dimensional View

What Is Convex Optimization?
Introduction
Basic concepts
Smooth Convex Optimization

Tools for Convex Optimization
Introduction
Convex Sets

Convex Functions
Subdifferential Calculus
Conjugate Functions
ε-Subdifferential
Epigraphical Properties of Conjugate Functions

Basic Optimality Conditions using the Normal Cone
Introduction
Slater Constraint Qualification
Abadie Constraint Qualification
Convex Problems with Abstract Constraints
Max-Function Approach
Cone-Constrained Convex Programming

Saddle Points, Optimality, and Duality
Introduction
Basic Saddle Point Theorem
Affine Inequalities and Equalities and Saddle Point Condition
Lagrangian Duality
Fenchel Duality
Equivalence between Lagrangian and Fenchel Duality: Magnanti’s Approach

Enhanced Fritz John Optimality Conditions
Introduction
Enhanced Fritz John Conditions Using the Subdifferential
Enhanced Fritz John Conditions under Restrictions
Enhanced Fritz John Conditions in the Absence of Optimal Solution
Enhanced Dual Fritz John Optimality Conditions

Optimality without Constraint Qualification
Introduction
Geometric Optimality Condition: Smooth Case
Geometric Optimality Condition: Nonsmooth Case
Separable Sublinear Case

Sequential Optimality Conditions and Generalized Constraint Qualification
Introduction
Sequential Optimality: Thibault’s Approach
Fenchel Conjugates and Constraint Qualification
Applications to Bilevel Programming Problems

Representation of the Feasible Set and KKT Conditions
Introduction
Smooth Case
Nonsmooth Case

Weak Sharp Minima in Convex Optimization
Introduction
Weak Sharp Minima and Optimality

Approximate Optimality Conditions
Introduction
ε-Subdifferential Approach
Max-Function Approach
ε-Saddle Point Approach
Exact Penalization Approach
Ekeland’s Variational Principle Approach
Modified ε-KKT Conditions
Duality-Based Approach to ε-Optimality

Convex Semi-Infinite Optimization
Introduction
Sup-Function Approach
Reduction Approach
Lagrangian Regular Point
Farkas–Minkowski Linearization
Noncompact Scenario: An Alternate Approach

Convexity in Nonconvex Optimization
Introduction
Maximization of a Convex Function
Minimization of d.c. Functions

Bibliography
Index