Table of Contents
1 Cover
6 1 Optimization Problems with Differentiable Objective Functions 1.1. Basic concepts 1.2. Optimization problems with objective functions of one variable 1.3. Optimization problems with objective functions of several variables 1.4. Constrained optimization problems 1.5. Exercises
7 2 Convex Sets 2.1. Convex sets: basic definitions 2.2. Combinations of points and hulls of sets 2.3. Topological properties of convex sets 2.4. Theorems on separation planes and their applications 2.5. Systems of linear inequalities and equations 2.6. Extreme points of a convex set 2.7. Exercises
8 3 Convex Functions 3.1. Convex functions: basic definitions 3.2. Operations in the class of convex functions 3.3. Criteria of convexity of differentiable functions 3.4. Continuity and differentiability of convex functions 3.5. Convex minimization problem 3.6. Theorem on boundedness of Lebesgue set of a strongly convex function 3.7. Conjugate function 3.8. Basic properties of conjugate functions 3.9. Exercises
9 4 Generalizations of Convex Functions 4.1. Quasi-convex functions 4.2. Pseudo-convex functions 4.3. Logarithmically convex functions 4.4. Convexity in relation to order 4.5. Exercises
10 5 Sub-gradient and Sub-differential of Finite Convex Function 5.1. Concepts of sub-gradient and sub-differential 5.2. Properties of sub-differential of convex function 5.3. Sub-differential mapping 5.4. Calculus rules for sub-differentials 5.5. Systems of convex and linear inequalities 5.6. Exercises
11 6 Constrained Optimization Problems 6.1. Differential conditions of optimality 6.2. Sub-differential conditions of optimality 6.3. Exercises 6.4. Constrained optimization problems 6.5. Exercises 6.6. Dual problems in convex optimization 6.7. Exercises
12 Solutions, Answers and Hints
13 References
14 Index
List of Illustrations
1 Chapter 1Figure 1.1. Example 1.5Figure 1.2. Example 1.6
2 Chapter 2Figure 2.1. Convex set X1. Non-convex set X2Figure 2.2. X1 is a cone. X2 is a convex coneFigure 2.3. Conjugate conesFigure 2.4. Affine set and linear subspaceFigure 2.5. a) Convex hull. b) Conic hullFigure 2.6. a) Convex polyhedron. b) Polyhedral coneFigure 2.7. Unbounded closed convex setFigure 2.8. Projection of a point onto a setFigure 2.9. Sets X1 and X2 are: a) properly separated; b) strongly separated; c)...Figure 2.10. a), c) Properly supporting hyperplanes; b) supporting hyperplane
3 Chapter 3Figure 3.1. Convex functionFigure 3.2. Epigraph of convex functionFigure 3.3. Epigraph
