Basic Mathematical Programming Theory

ISBN-13: 9783031303234 / Angielski / Twarda / 2023

Giorgio Giorgi;Bienvenido Jiménez;Vicente Novo

Opis

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.

Spis treści

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 Optimiza-

tion 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

O autorze

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.

Blurb

This book presents a unified, progressive treatment of the basic mathematical tools of 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 in Rn 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.