Numerical Methods of Mathematical Optimization

Numerical Methods of Mathematical Optimization: With ALGOL and FORTRAN Programs reviews the theory and the practical application of the numerical methods of mathematical optimization. An ALGOL and a FORTRAN program was developed for each one of the algorithms described in the theoretical section. This should result in easy access to the application of the different optimization methods. Comprised of four chapters, this volume begins with a discussion on the theory of linear and nonlinear optimization, with the main stress on an easily understood, mathematically precise presentation. In addition to the theoretical considerations, several algorithms of importance to the numerical application of optimization theory are described. The next chapter explains the computer programs used in actual optimization, which have the form of procedures or subroutines. The book concludes with an analysis of ALGOL and FORTRAN, paying particular attention to their use in global optimization procedures as well as for the simplex and duoplex methods and the decomposition, Gomory, Beale, and Wolfe algorithms. This monograph will be helpful to students and practitioners of computer science and applied mathematics.

Preface to the German Edition

1. Linear Optimization

1.1 General Formulation of Linear Optimization

1.2 The Simplex Method

1.3 Determination of a Feasible Initial Solution

1.4 The Fundamental Theorem and the Simplex Criterion of the Linear Optimization Theory

1.5 Degeneracies

1.6 Dual Linear Optimization Problems

1.7 The Dual Simplex Method

1.8 The Revised Simplex Method

1.9 The Decomposition Algorithm

1.10 The Duoplex Algorithm

1.11 Linear Integer Optimization

2. Nonlinear Optimization

2.1 Convex Domains and Functions

2.2 General Nonlinear Optimization

2.3 Convex Optimization

2.4 The Kuhn-Tucker Conditions

2.5 Quadratic Optimization

2.6 Duality in the Case of Quadratic Optimization

2.7 The Method of Beale

2.8 The Method of Wolfe

2.9 A Look at Further Methods

3. Explanations of the Computer Programs

3.1 The Subroutine System

3.2 The Use of the Optimization Programs

3.3 Numerical Properties

3.4 General Discussion of Variable Notations and Storage Organization

3.5 Properties of Individual Optimization Programs

4. ALGOL and FORTRAN Programs

4.1 Global Procedures

4.2 ALGOL Program for the Simplex Method

4.3 FORTRAN Program for the Simplex Method

4.4 ALGOL Program for the Dual Simplex Method

4.5 FORTRAN Program for the Dual Simplex Method

4.6 ALGOL Program for the Revised Simplex Method

4.7 FORTRAN Program for the Revised Simplex Method

4.8 ALGOL Program for the Decomposition Algorithm

4.9 FORTRAN Program for the Decomposition Algorithm

4.10 ALGOL Program for the Duoplex Method

4.11 FORTRAN Program for the Duoplex Method

4.12 ALGOL Program for the Gomory Algorithm

4.13 FORTRAN Program for the Gomory Algorithm

4.14 ALGOL Program for the Beale Algorithm

4.15 FORTRAN Program for the Beale Algorithm

4.16 ALGOL Program for the Wolfe Algorithm

4.17 FORTRAN Program for the Wolfe Algorithm

List of Existing Computer Programs

Bibliography

Addendum: Version of Computer Programs for Practical Application

Index