Skip to main content
SpringerLink
Log in

Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value

  • Published:
Mathematical Programming Submit manuscript

Abstract

Variants of Karmarkar's algorithm are given for solving linear programs with unknown optimal objective valuez *. These new methods combine the approach of Goldfarb and Mehrotra for relaxing the requirement that certain projections be computed exactly with the approach of Todd and Burrell for generating an improving sequence of lower bounds forz * using dual feasible solutions. These methods retain the polynomial-time complexity of Karmarkar's algorithm.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. K.M. Anstreicher, A monotonic projective algorithm for fractional linear programming,Algorithmica 1 (1986) 483–498.

    Google Scholar 

  2. D.M. Gay, “A variant of Karmarkar's linear programming algorithm for problems in standard form,”Mathematical Programming 37 (1987) 81–90.

    Google Scholar 

  3. D. Goldfarb and S. Mehrotra, “A relaxed version of Karmarkar's method,”Mathematical Programming 40 (1988) (next issue).

  4. C. Gonzaga, “A conical projection algorithm for linear programming,” Department of Electrical Engineering and Computer Science, University of California (Berkeley, CA, 1985).

    Google Scholar 

  5. N. Karmarkar, “A new polynomial-time algorithm for linear programming,”Combinatorica 4 (1984) 373–395.

    Google Scholar 

  6. S. Mehrotra, “Variants of Karmarkar's algorithm: Theoretical complexity and practical implementation,” Ph.D. Thesis, Department of Industrial Engineering and Operations Research (Columbia University, NY, 1987).

    Google Scholar 

  7. C.C. Paige and M.A. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,”ACM Transactions on Mathematical Software 8 (1982) 43–71.

    Google Scholar 

  8. P.F. Pickel, “Approximate projections for the Karmarkar algorithm,” Manuscript, Polytechnic Institute of New York (Farmingdale, NY, 1985).

    Google Scholar 

  9. M.J. Todd and B.P. Burrell (1986), “An extension of Karmarkar's algorithm for linear programming using dual variables,”Algorithmica 1 (1986) 409–424.

    Google Scholar 

  10. J.A. Tomlin, “An experimental approach to Karmarkar's projective method for linear programming,”Mathematical Programming Study 31 (1987) 175–191.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Columbia University, 10027, New York, NY, USA

    Donald Goldfarb

  2. Department of Industrial Engineering and Management Studies, Northwestern University, 60208, Evanston, IL, USA

    Sanjay Mehrotra

Authors
  1. Donald Goldfarb
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sanjay Mehrotra
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported in part by NSF Grants DMS-85-12277 and CDR-84-21402, and ONR Contract N0014-87-K0214.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldfarb, D., Mehrotra, S. Relaxed variants of Karmarkar's algorithm for linear programs with unknown optimal objective value. Mathematical Programming 40, 183–195 (1988). https://doi.org/10.1007/BF01580729

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01580729

Key words

Search

Navigation