Linear Optimization and Approximation

1. Front Matter

   Pages N2-ix

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2. Introduction and Preliminaries

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 1-19

3. Weak Duality

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 20-36

4. Applications of Weak Duality in Uniform Approximation

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 37-57

5. Duality Theory

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 58-91

6. The Simplex Algorithm

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 92-114

7. Numerical Realization of the Simplex Algorithm

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 115-133

8. A General Three-Phase Algorithm

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 134-152

9. Approximation Problems by Chebyshev Systems

   • Klaus Glashoff, Sven-Åke Gustafson

   Pages 153-174

10. Examples and Applications of Semi-Infinite Programming

    • Klaus Glashoff, Sven-Åke Gustafson

    Pages 175-192

11. Back Matter

    Pages 193-198

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A linear optimization problem is the task of minimizing a linear real-valued function of finitely many variables subject to linear con­ straints; in general there may be infinitely many constraints. This book is devoted to such problems. Their mathematical properties are investi­ gated and algorithms for their computational solution are presented. Applications are discussed in detail. Linear optimization problems are encountered in many areas of appli­ cations. They have therefore been subject to mathematical analysis for a long time. We mention here only two classical topics from this area: the so-called uniform approximation of functions which was used as a mathematical tool by Chebyshev in 1853 when he set out to design a crane, and the theory of systems of linear inequalities which has already been studied by Fourier in 1823. We will not treat the historical development of the theory of linear optimization in detail. However, we point out that the decisive break­ through occurred in the middle of this century. It was urged on by the need to solve complicated decision problems where the optimal deployment of military and civilian resources had to be determined. The availability of electronic computers also played an important role. The principal computational scheme for the solution of linear optimization problems, the simplex algorithm, was established by Dantzig about 1950. In addi­ tion, the fundamental theorems on such problems were rapidly developed, based on earlier published results on the properties of systems of linear inequalities.

• Approximation
• Dualität (Math.)
• Lineare Optimierung
• algorithms
• calculus
• linear optimization
• optimization

• Institut für Angewandte Mathematik, Universität Hamburg, Federal Republic of Germany

  Klaus Glashoff

• Department of Numerical Analysis and Computing Sciences, Royal Institute of Technology, Stockholm 70, Sweden

  Sven-Åke Gustafson

• Centre for Mathematical Analysis, Australian National University, Canberra, Australia

  Sven-Åke Gustafson

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