Basic Mathematical Programming Theory

The subject of (static) optimization, also called mathematical programming, is one of the most important and widespread branches of modern mathematics, serving as a cornerstone of such scientific subjects as economic analysis, operations research, management sciences, engineering, chemistry, physics, statistics, computer science, biology, and social sciences. This book presents a unified, progressive treatment of the basic mathematical tools of mathematical programming theory. The authors expose said tools, along with results concerning the most common mathematical programming problems formulated in a finite-dimensional setting, forming the basis for further study of the basic questions on the various algorithmic methods and the most important particular applications of mathematical programming problems. This book assumes no previous experience in optimization theory, and the treatment of the various topics is largely self-contained. Prerequisites are the basic tools of differential calculus for functions of several variables, the basic notions of topology and of linear algebra, and the basic mathematical notions and theoretical background used in analyzing optimization problems. The book is aimed at both undergraduate and postgraduate students interested in mathematical programming problems but also those professionals who use optimization methods and wish to learn the more theoretical aspects of these questions.

Puts forth a modern approach to the most valuable mathematical tools within mathematical programming theory Forms the basis for further study of various algorithmic methods and applications of mathematical programming problems Aids courses dedicated to optimization, economic analysis, and others related to optimization theory and its methods

Autorentext

Prof. Giorgio Giorgi teaches Mathematics at the Faculty of Economics of the University of Pavia. His research interests essentially focus on mathematical economics, generalized convexity, and optimization.

Bienvenido Jiménez and Vicente Novo are professors of Applied Mathematics at the National University of Distance Education, Madrid, Spain. Their research focus on smooth and nonsmooth optimization, mathematical programming and multiobjective, vector and set optimization.

Inhalt
Preface.- Chapter 1. Basic Notions and Definitions.- 1.1. Introduction.- 1.2. Basic Notions of Analysis and Linear Algebra.- 1.3. Basic Notions and Properties of Optimization Problems.- Chapter 2. Elements of Convex Analysis. Theorems of the Alternative for Linear Systems. Tangent Cones.- 2.1. Elements of Convex Analysis.- 2.2. Theorems of the Alternative for Linear Systems.- 2.3. Tangent Cones.- Chapter 3. Convex Functions and Generalized Convex Functions.- 3.1. Convex Functions.- 3.2. Generalized Convex Functions.- 3.3. Optimality Properties of Convex and Generalized Convex Functions. Theorems of the Alternative for Nonlinear Systems.- Chapter 4. Unconstrained Optimization Problems. Set-Constrained Optimization Problems. Classical Constrained Optimization Problems.- 4.1. Unconstrained Optimization Problems.- 4.2. Set-Constrained Optimization Problems.- 4.3. Optimization Problems with Equality Constraints (Classical Constrained Optimization Problems).- Chapter 5. Constrained Optimization Problems with Inequality Constraints.- 5.1. First-Order Conditions.- 5.2. Constraint Qualifications.- 5.3. Second-Order Conditions.- 5.4. Other Formulations of the Problem. Some Examples.- Chapter 6. Constrained Optimization Problems with Mixed Constraints.- 6.1. First-Order Conditions.- 6.2. Constraint Qualifications.- 6.3. Second-Order Conditions.- 6.4. Problems with a Set Constraint. Asymptotic Optimality Conditions.- Chapter 7.Sensitivity Analysis.- 7.1. General Results.- 7.2. Sensitivity Results for Right-Hand Side Perturbations.- Chapter 8. Convex Optimization: Saddle Points Characterization and Introduction to Duality.- 8.1. Convex Optimization: Saddle Points Characterization.- 8.2. Introduction to Duality.- Chapter 9. Linear Programming and Quadratic Programming.- 9.1. Linear Programming.- 9.2. Duality for Linear Programming.- 9.3. Quadratic Programming.- Chapter 10. Introduction to Nonsmooth Optimization Problems.- 10.1. The Convex Case.- 10.2. The Lipschitz Case.- 10.3.The Axiomatic Approach of K.-H. Elster and J. Thierfelder to Nonsmooth Optimization.- Chapter 11. Introduction to Multiobjective Optimization.- 11.1. Optimality Notions.- 11.2. The Weighted Sum Method and Optimality Conditions.- References.- Index.