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This book provides easy access to the basic principles and methods for solving constrained and unconstrained convex optimization problems. Included are sections that cover: basic methods for solving constrained and unconstrained optimization problems with differentiable objective functions; convex sets and their properties; convex functions and their properties and generalizations; and basic principles of sub-differential calculus and convex programming problems.
Cover -- Half-Title Page -- Title Page -- Copyright Page -- Contents -- Notations -- Introduction -- 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 -- 1.4.1. Problems with equality constraints -- 1.4.2. Problems with equality and inequality constraints -- 1.5. Exercises -- 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 -- 2.4.1. Projection of a point onto a set -- 2.4.2. Separation of two sets -- 2.5. Systems of linear inequalities and equations -- 2.6. Extreme points of a convex set -- 2.7. Exercises -- 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 -- 4. Generalizations of Convex Functions -- 4.1. Quasi-convex functions -- 4.1.1. Differentiable quasi-convex functions -- 4.1.2. Operations that preserve quasi-convexity -- 4.1.3. Representation in the form of a family of convex functions. -- 4.1.4. The maximization problem for quasi-convex functions -- 4.1.5. Strictly quasi-convex functions -- 4.1.6. Strongly quasi-convex functions
4.2. Pseudo-convex functions -- 4.3. Logarithmically convex functions -- 4.3.1. Properties of logarithmically convex functions -- 4.3.2. Integration of logarithmically concave functions -- 4.4. Convexity in relation to order -- 4.5. Exercises -- 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
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 -- 6.4.1. Principle of indeterminate Lagrange multipliers -- 6.4.2. Differential form of the Kuhn-Tucker theorem -- 6.4.3. Second-order conditions of optimality -- 6.5. Exercises -- 6.6. Dual problems in convex optimization -- 6.6.1. Kuhn-Tucker vector -- 6.6.2. Dual optimization problems -- 6.6.3. Kuhn-Tucker theorem for non-differentiable functions -- 6.6.4. Method of perturbations
6.6.5. Economic interpretations of the Kuhn-Tucker vector.
Includes bibliographical references and index.
Convex Optimization : Introductory Course
Newark : John Wiley & Sons, Incorporated, ©2021