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A globally and quadratically convergent affine scaling method for linearℓ 1 problems

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Abstract

Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this “interior point” philosophy can be adapted to the linear ℓ1 problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.

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References

  1. E. Barnes, “A variation on Karmarkar's algorithm for solving linear programming problems,”Mathematical Programming 36 (1986) 174–182.

    Google Scholar 

  2. I. Barrodale and F. Roberts, “An improved algorithm for discrete ℓ1 linear approximation,”SIAM Journal on Numerical Analysis 10 (1972 839–848.

    Google Scholar 

  3. R.H. Bartels, A.R. Conn and J.W. Sinclair, “Minimization techniques for piecewise differentiable functions: the ℓ1 solution to an overdetermined linear system,”SIAM Journal Numerical Analysis 15 (1978) 224–240.

    Google Scholar 

  4. C. Bischof, “QR factorization algorithms for coarse-grained distributed systems,” Technical Report 88-939, Cornell University (Ithaca, NY, 1988).

    Google Scholar 

  5. T.F. Coleman and P. Plassmann, “Solution of nonlinear least-squares problems on a multiprocessor,” in: G. van Zee and J. van de Vorst, eds.,Parallel Computing 1988, Shell Conference Proceedings, Lecture Notes in Computer Science No. 384 (Springer, Berlin, 1989).

    Google Scholar 

  6. T.F. Coleman and A. Pothen, “The null space problem I: Complexity,”SIAM Journal on Algebraic and Discrete Methods 7 (1987) 527–537.

    Google Scholar 

  7. T.F. Coleman and A. Pothen, “The null space problem II: Algorithms,”SIAM Journal on Algebraic and Discrete Methods 8 (1987) 544–563.

    Google Scholar 

  8. J.E. Dennis, Jr. and J.J. Moré, “Quasi-Newton methods, motivation and theory,”SIAM Review 19 (1977) 46–89.

    Google Scholar 

  9. I. Dikin, “Iterative solution of problems of linear and quadratic programming,”Doklady Akademiia Nauk SSSR 174 (1967) 747–748.

    Google Scholar 

  10. P. Gill, W. Murray, M. Saunders, J. Tomlin and M. Wright, “On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method,”Mathematical Programming 36 (1986) 183–209.

    Google Scholar 

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

    Google Scholar 

  12. M.S. Meketon, “Least absolute value regression,” Technical Report, AT&T Bell Laboratory (Murray Hill, NJ, 1987).

    Google Scholar 

  13. C.B. Moler, J. Little, S. Bangert and S. Kleiman,ProMatlab User's Guide (MathWorks, Sherborn, MA, 1987).

    Google Scholar 

  14. J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).

    Google Scholar 

  15. M.R. Osborne,Finite Algorithms in Optimization and Data Analysis (Wiley, New York, 1985).

    Google Scholar 

  16. J. Renegar, “A polynomial-time algorithm, based on Newton's method, for linear programming,”Mathematical Programming 40 (1988) 59–93.

    Google Scholar 

  17. S.A. Ruzinsky and E.T. Olsen, “ℓ1 and ℓ∞ minimization via a variant of Karmarkar's algorithm,”IEEE Transactions on Acoustics Speech and Signal Processing 37 (1989) 245–253.

    Google Scholar 

  18. E. Seneta and W.L. Steiger, “A new lad curve-fitting algorithm: Slightly overdetermined equation systems in ℓ1,”Discrete Applied Mathematics 7 (1984) 79–91.

    Google Scholar 

  19. M. Todd, “Polynomial algorithms for linear programming,” in: H. Eiselt and G. Pederzoli, eds.,Advances in Optimization and Control (Springer, Berlin, 1988) pp. 49–66.

    Google Scholar 

  20. C. Van Loan, “On the method of weighting for equality-constrained least squares problems,”SIAM Journal on Numerical Analysis 22 (1985) 851–864.

    Google Scholar 

  21. R.J. Vanderbei, M.S. Meketon and B.A. Freedman, “A modification of Karmarkar's linear programming algorithm,”Algorithmica 1 (1986) 395–407.

    Google Scholar 

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Authors and Affiliations

  1. Computer Science Department and Center for Applied Mathematics, Cornell University, 4130 Upson Hall, 14853, Ithaca, NY, USA

    Thomas F. Coleman & Yuying Li

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  1. Thomas F. Coleman
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  2. Yuying Li
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Additional information

Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University, and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.

Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.

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Coleman, T.F., Li, Y. A globally and quadratically convergent affine scaling method for linearℓ 1 problems. Mathematical Programming 56, 189–222 (1992). https://doi.org/10.1007/BF01580899

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  • DOI: https://doi.org/10.1007/BF01580899

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