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Homotopy techniques in linear programming

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Abstract

In this note, we consider the solution of a linear program, using suitably adapted homotopy techniques of nonlinear programming and equation solving that move through the interior of the polytope of feasible solutions. The homotopy is defined by means of a quadratic regularizing term in an appropriate metric. We also briefly discuss algorithmic implications and connections with the affine variant of Karmarkar's method.

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References

  1. G. B. Dantzig,Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963.

    MATH  Google Scholar 

  2. B. C. Eaves, A view of complementary pivot theory (or solving equations with homotopies), inConstructive Approaches to Mathematical Models (C. V. Coffman and G. J. Fix, eds.), Academic Press, New York, 1979, pp. 153–170.

    Google Scholar 

  3. B. C. Eaves and H. Scarf, The solution of systems of piecewise-linear equations,Math. Oper. Res.,1 (1976), 1–27.

    MATH  MathSciNet  Google Scholar 

  4. C. B. Garcia and W. I. Zangwill,Pathways to Solutions, Fixed Points and Equilibria, Prentice-Hall, Englewood Cliffs, NJ, 1981.

    MATH  Google Scholar 

  5. P. E. Gill, W. Murray, M. A. Saunders, J. A. Tomlin, and M. H. Wright, On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method, Technical Report SOL 85-11, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, CA, 1985.

    Google Scholar 

  6. M. Hirsch and S. Smale, On algorithms for solvingf(x) = 0,Comm. Pure Appl. Math.,32 (1979), 281–312.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Karmarkar, A new polynomial-time algorithm for linear programming,Proceedings of the 16th Annual ACM Symposium on Theory of Computing, 1984, pp. 302–311.

  8. L. G. Khachiyan, A polynomial algorithm for linear programming,Dokl. Akad. Nauk SSSR,244 (1979), 1093–1096 (translated inSoviet Math. Dokl,20 (1979), 191–194).

    MATH  MathSciNet  Google Scholar 

  9. C. E. Lemke, On complementary pivot theory, inMathematics of the Decision Sciences (G. B. Dantzig and A. F. Veinott, eds.), American Mathematical Society, Providence, RI, 1968.

    Google Scholar 

  10. O. L. Mangasarian, Iterative solution of linear programs,SIAM J. Numer. Anal,18 (1981), 606–614.

    Article  MATH  MathSciNet  Google Scholar 

  11. N. Megiddo, A variation on Karmarkar's algorithm, Preliminary Report, IBM Research Laboratory, San Jose, CA, 1985.

    Google Scholar 

  12. B. N. Pschenichny and Y. M. Danilin,Numerical Methods in Extremum Problems, M.I.R., Moscow, 1975 (English translation, 1978).

    Google Scholar 

  13. R. T. Rockafellar, Augmented lagrangians and applications of the proximal point algorithm in convex programming,Math. Oper. Res.,1 (1976), 97–116.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. F. Shanno, A reduced gradient variant of Karmarkar's algorithm, Working Paper 85-01, Graduate School of Administration, University of California, Davis, CA, 1985.

    Google Scholar 

  15. N. Z. Shor, Generalized gradient methods of non-differentiable optimization employing space dilation operations, inMathematical Programming: The State of the Art (A. Bachem, M. Grotschel, and B. Korte, eds.), Springer-Verlag, Berlin, 1983, pp. 501–529.

    Google Scholar 

  16. S. Smale, A convergent process of price adjustment and global Newton methods,J. Math. Econom.,3 (1976), 107–120.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Smale, The problem of the average speed of the simplex method, inMathematical Programming: The State of the Art (A. Bachem, M. Grotschel, and B. Korte, eds.), Springer-Verlag, Berlin, 1983, pp. 530–539.

    Google Scholar 

  18. G. Strang, Karmarkar's algorithm in a nutshell,SIAM News,18 (1985), 13.

    Google Scholar 

  19. M. J. Todd and B. P. Burrell, An extension of Karmarkar's algorithm for linear programming using dual variables, Technical Report 648, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 1985.

    Google Scholar 

  20. J. A. Tomlin, An experimental approach to Karmarkar's linear programming algorithm, Preprint, Ketron Inc., Mountain View, CA, 1985.

    Google Scholar 

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Authors and Affiliations

  1. C.D.S.S., P.O. Box 4908, 94704, Berkeley, CA, USA

    J. L. Nazareth

  2. IIASA, A-2361, Laxenburg, Austria

    J. L. Nazareth

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  1. J. L. Nazareth
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Additional information

Communicated by Nimrod Megiddo.

This is a revised version of a paper previously entitled “Karmarkar's Method and Homotopies with Restarts”.

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Nazareth, J.L. Homotopy techniques in linear programming. Algorithmica 1, 529–535 (1986). https://doi.org/10.1007/BF01840461

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  • DOI: https://doi.org/10.1007/BF01840461

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