Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Surrogate methods for linear inequalities

Abstract

Three algorithms are developed and validated for finding a pointx inR n that satisfies a given system of inequalities,Ax≤b. A andb are a given matrix and a given vector inR m×n andR m, respectively, with the rows ofA assumed normalized. The algorithms are iterative and are based upon the orthogonal projection of an infeasible point onto the manifold of the bounding hyperplanes of some of the given constraints. The choice of the active constraints and the actual projection are accomplished through the use of surrogate constraints.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Agmon, S.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 382–392, 1954.

  2. 2.

    Motzkin, T. S., andSchoenberg, I. J.,The Relaxation Method for Linear Inequalities, Canadian Journal of Mathematics, Vol. 6, pp. 393–404, 1954.

  3. 3.

    Jeroslow, R. G.,Some Relaxation Methods for Linear Inequalities, Cahiers du CERO, Vol. 21, pp. 43–53, 1979.

  4. 4.

    Goffin, J. L.,The Relaxation Method for Solving Systems of Linear Inequalities, Mathematics of Operations Research, Vol. 5, pp. 388–414, 1980.

  5. 5.

    Todd, M. J.,Some Remarks on the Relaxation Method for Linear Inequalities, Technical Report No. 419, Cornell University, 1979.

  6. 6.

    Goldfarb, D., andTodd, M. J.,Modifications and Implementation of the Short Khachian Algorithm for Linear Programming, Technical Report No. 446, Cornell University, 1980.

  7. 7.

    Krol, Y., andMirman, B.,Some Practical Modifications of Ellipsoid Method for LP Problems, ARCON (undated report).

  8. 8.

    Hoffman, A. J.,On Approximate Solutions of Systems of Linear Inequalities, Journal of Research of the National Bureau of Standards, Vol. 49, pp. 263–265, 1952.

Download references

Author information

Additional information

This work was supported in part by the City University of New York Research Center. The author thanks Professor D. Goldfarb for suggesting the problem and also for his valuable literature information and time. The word surrogate was borrowed from one of his works.

Communicated by F. Zirilli

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Oko, S.O. Surrogate methods for linear inequalities. J Optim Theory Appl 72, 247–268 (1992). https://doi.org/10.1007/BF00940518

Download citation

Key Words

  • Surrogate constraints
  • hyperplanes
  • halfspaces
  • orthogonal projections
  • manifolds
  • active constraints
  • linear combinations
  • outward normals