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A combined phase I—phase II scaled potential algorithm for linear programming

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Abstract

We develop an extension of the affinely scaled potential reduction algorithm which simultaneously obtains feasibility and optimality in a standard form linear program, without the addition of any “M” terms. The method, and its lower-bounding procedure, are particularly simple compared with previous interior algorithms not requiring feasibility.

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References

  • K.M. Anstreicher, “Analysis of a modified Karmarkar algorithm for linear programming,” Technical Report Series B #84, Yale School of Management (New Haven, CT, 1985).

    Google Scholar 

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

    Google Scholar 

  • K.M. Anstreicher, “A combined phase I-phase II projective algorithm for linear programming,”Mathematical Programming 43 (1989) 209–223.

    Google Scholar 

  • G. de Ghellinck and J.-P. Vial, “A polynomial Newton method for linear programming,”Algorithmica 1 (1986) 425–453.

    Google Scholar 

  • C. Fraley, “Linear updates for a single-phase projective method,” COMIN, University of Geneva (Geneva, Switzerland, 1989).

    Google Scholar 

  • C. Fraley and J.-P. Vial, “Single-phase versus multi-phase projective methods for linear programming,” COMIN, University of Geneva (Geneva, Switzerland, 1989).

    Google Scholar 

  • R.M. Freund, “Projective transformations for interior point methods, part I: basic theory and linear programming,” Working Paper OR 179-88, OR Center, MIT (Cambridge, MA, 1988).

    Google Scholar 

  • R.M. Freund, “Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function,”Mathematical Programming 51 (1991a) 203–222.

    Google Scholar 

  • R.M. Freund, “A potential-function reduction algorithm for solving a linear program directly from an infeasible “warm start”,”Mathematical Programming (Series B) 52 (1991b) 441–466, this issue.

    Google Scholar 

  • R.M. Freund, “Theoretical efficiency of a shifted barrier function algorithm for linear programming,”Linear Algebra and its Applications 152 (1991c) 19–41.

    Google Scholar 

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

    Google Scholar 

  • C.C. Gonzaga, “Conical projection algorithms for linear programming,”Mathematical Programming 43 (1989) 151–173.

    Google Scholar 

  • C.C. Gonzaga, “Polynominal affine algorithms for linear programming,”Mathematical Programming 49 (1990) 7–21.

    Google Scholar 

  • C.C. Gonzaga, “Large-step path-following methods for linear programming, Part II: potential reduction method,”SIAM Journal on Optimization 1 (1991) 280–292.

    Google Scholar 

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

    Google Scholar 

  • A.E. Steger, “An extension of Karmarkar's algorithm for bounded linear programming problems,” M.S. Thesis, State University of New York (Stonybrook, NY, 1985).

    Google Scholar 

  • M.J. Todd, “The effect of sparsity, degeneracy, and null and unbounded variables on variants of Karmarkar's linear programming algorithm,” Technical Report 857, School of OR/IE, Cornell University (Ithaca, NY, 1989).

    Google Scholar 

  • M.J. Todd, “On Anstreicher's combined phase I-phase II projective algorithm for linear programming,” to appear in:Mathematical Programming (1992).

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

    Google Scholar 

  • M.J. Todd and Y. Wang, “On combined phase I-phase II projective methods for linear programming,” Technical Report 877, School of OR/IE, Cornell University (Ithaca, NY, 1989).

    Google Scholar 

  • Y. Ye, “An O(n 3 L) potential reduction algorithm for linear programming,”Mathematical Programming 50 (1991) 239–258.

    Google Scholar 

  • Y. Ye and M. Kojima, “Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming,”Mathematical Programming 39 (1987) 305–317.

    Google Scholar 

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Authors and Affiliations

  1. Department of Management Sciences, University of Iowa, 52242, Iowa City, IA, USA

    Kurt M. Anstreicher

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  1. Kurt M. Anstreicher
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Additional information

This paper was written while the author was a research fellow at the Center for Operations Reasearch and Econometrics, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.

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Anstreicher, K.M. A combined phase I—phase II scaled potential algorithm for linear programming. Mathematical Programming 52, 429–439 (1991). https://doi.org/10.1007/BF01582899

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  • DOI: https://doi.org/10.1007/BF01582899

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