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A least-distance programming procedure for minimization problems under linear constraints

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Abstract

In this paper, an algorithm is developed for solving a nonlinear programming problem with linear contraints. The algorithm performs two major computations. First, the search vector is determined by projecting the negative gradient of the objective function on a polyhedral set defined in terms of the gradients of the equality constraints and the near binding inequality constraints. This least-distance program is solved by Lemke's complementary pivoting algorithm after eliminating the equality constraints using Cholesky's factorization. The second major calculation determines a stepsize by first computing an estimate based on quadratic approximation of the function and then finalizing the stepsize using Armijo's inexact line search.

It is shown that any accumulation point of the algorithm is a Kuhn-Tucker point. Furthermore, it is shown that, if an accumulation point satisfies the second-order sufficiency optimality conditions, then the whole sequence of iterates converges to that point. Computational testing of the algorithm is presented.

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Author notes

  1. M. S. Bazaraa (Professor)

    Present address: Burnham Service Corporation, Columbus, Georgia

Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia

    M. S. Bazaraa (Professor)

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia

    J. J. Goode (Professor)

Authors
  1. M. S. Bazaraa
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  2. J. J. Goode
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Additional information

Communicated by D. G. Luenberger

The work of the first author was supported under AFOSR Grant No. AFOSR-80-0195.

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Bazaraa, M.S., Goode, J.J. A least-distance programming procedure for minimization problems under linear constraints. J Optim Theory Appl 40, 489–514 (1983). https://doi.org/10.1007/BF00933968

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