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E-raamat: Basic Mathematical Programming Theory

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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.

Arvustused

This book serves as a comprehensive and indispensable tool for Ph.D. students and young researchers in the Field of nonlinear programming. it offers a substantial coverage of theoretical results concerning optimality conditions in nonlinear programming. Certain chapters are particularly valuable as teaching materials for advanced master's courses. (Giovanni Fasano, Mathematical Reviews, Decemeber, 2024)

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.
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.
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