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Cubically convergent method for locating a nearby vertex in linear programming

Abstract

Given a point sufficiently close to a nondegenerate basic feasible solutionx* of a linear program, we show how to generate a sequence {p k} that converges to the 0–1 vector sign(x*) at aQ-cubic rate. This extremely fast convergence enables us to determine, with a high degree of certainty, which variables will be zero and which will be nonzero at optimality and then constructx* from this information.

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References

  1. 1.

    Tapia, R. A., andZhang, Y.,A Fast Optimal Basis Identification Technique for Interior Point Linear Programming Methods, Technical Report No. 89-1, Department of Mathematical Sciences, Rice University, Houston, Texas, 1989.

  2. 2.

    Karmarkar, N.,A New Polynomial Time Algorithm for Linear Programming, Combinatorica, Vol. 4, pp. 373–395, 1984.

  3. 3.

    Barnes, E. R.,A Variation of Karmarkar's Algorithm for Solving Linear Programming Problems, Mathematical programming, Vol. 36, pp. 174–182, 1986.

  4. 4.

    Dikin, I. I.,Iterative Solution of Problems of Linear and Quadratic Programming, Soviet Mathematics Doklady, Vol. 8, pp. 674–675, 1967.

  5. 5.

    Vanderbei, R. J., Meketon, M. S., andFreeman, B. A.,On a Modification of Karmarkar's Linear Progrzmming Algorithm, Algorithmica, Vol. 1, pp. 395–407, 1986.

  6. 6.

    Adler, I., Karmarkar, N., Resende, M. G. C., andVeiga, G.,An Implementation of Karmarkar's Algorithm for Linear Programming, Technical Report No. 86-8, Operations Research Center, University of California, Berkeley, California, 1986.

  7. 7.

    Monma, C. L., andMorton, A. J.,Computational Experience with a Dual Affine Variant of Karmarkar's Method for Linear Programming, OR Letters, Vol. 6, pp. 261–267, 1987.

  8. 8.

    Kojima, M., Mizuno, S., andYoshise, A. A Primal-Dual Interior Point Method for Linear Programming, Technical Report No. B-188, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1987.

  9. 9.

    Ye, Y., andTodd, M. J.,Containing and Shrinking Ellipsoids in the Path-Following Algorithm, Mathematical Programming (to appear).

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Additional information

This research was supported in part by NSF Cooperative Agreement No. CCR-8809615, by AFOSR Grant No. 89-0363, and by DOE Grant No. DEFG05-86ER 25017. The authors would like to thank Bob Bixby for helpful discussions.

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Tapia, R.A., Zhang, Y. Cubically convergent method for locating a nearby vertex in linear programming. J Optim Theory Appl 67, 217–225 (1990). https://doi.org/10.1007/BF00940473

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Key Words

  • Linear programming
  • vertex
  • Q-cubic convergence