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Surrogate methods for linear inequalities


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.

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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

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Oko, S.O. Surrogate methods for linear inequalities. J Optim Theory Appl 72, 247–268 (1992).

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Key Words

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