Linear Programming Part I: The Simplex Method


Linear programming (LP) problems occur in a diverse range of real-life applications in economic analysis and planning, operations research, computer science, medicine, and engineering. In such problems, it is known that any minima occur at the vertices of the feasible region and can be determined through a ‘brute-force’ or exhaustive approach by evaluating the objective function at all the vertices of the feasible region. However, the number of variables involved in a practical LP problem is often very large and an exhaustive approach would entail a considerable amount of computation. In 1947, Dantzig developed a method for the solution of LP problems known as the simplex method [1][2]. Although in the worst case, the simplex method is known to require an exponential number of iterations, for typical standard-form problems the number of iterations required is just a small multiple of the problem dimension [3]. For this reason, the simplex method has been the primary method for solving LP problems since its introduction.


Linear Programming Problem Feasible Region Simplex Method Feasible Point Active Constraint 
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  1. 1.
    G. B. Dantzig, “Programming in a linear structure,” Comptroller, USAF, Washington, D.C., Feb. 1948.Google Scholar
  2. 2.
    G. B. Dantzig, Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.MATHGoogle Scholar
  3. 3.
    P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization, vol. I, Addison-Wesley, Reading, 1991.MATHGoogle Scholar
  4. 4.
    R. Saigal, LP problem: A Modern Integrated Analysis, Kluwer Academic, Norwell, 1995.Google Scholar
  5. 5.
    G. H. Golub and C. F. Van Loan, Matrix Computation, 2nd ed., The Johns Hopkins University Press, Baltimore, 1989.Google Scholar
  6. 6.
    R. G. Bland, “Newfinite pivoting rules for the simplex method,” Math. Operations Research, vol. 2, pp. 103–108, May 1977.MATHMathSciNetGoogle Scholar
  7. 7.
    J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, Philadelphia, 1996.MATHGoogle Scholar
  8. 8.
    W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed., Cambridge University Press, Cambridge, UK, 1992.MATHGoogle Scholar
  9. 9.
    V. Klee and G. Minty, “How good is the simplex method?” in Inequalities, O. Shisha ed., pp. 159–175, Academic Press, New York, 1972.Google Scholar
  10. 10.
    M. H. Wright, “Interior methods for constrained optimization,” Acta Numerica, vol. 1, pp. 341–407, 1992.CrossRefGoogle Scholar

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