The ℓ1 solution of linear inequalities

Computational Statistics and Data Analysis

Volumn 50, Issue 1 SPEC. ISS., 2006, Pages 40-60

  Dax, Achiya ^a  

^a Hydrological Service of Israel   (Israel)

Author keywords

A new theorem of the alternative; Affine scaling; Duality; Inconsistent systems; Least absolute deviations; Linear inequalities

Indexed keywords

ALGORITHMS; NUMERICAL ANALYSIS; PROBLEM SOLVING; VECTORS;

A NEW THEOREM OF THE ALTERNATIVE; AFFINE SCALING; DUALITY; INCONSISTENT SYSTEMS; LEAST ABSOLUTE DEVIATIONS; LINEAR INEQUALITIES;

LINEAR SYSTEMS;

EID: 24644474737     PISSN: 01679473     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.csda.2004.07.007     Document Type: Article

References (46)

1. I. Adler, and R.D.C. Monteiro Limiting behaviour of the affine scaling continuous trajectories for linear programming problems Math. Programming 50 1991 29 51
2. I. Adler, N. Karmarkar, M.G.C. Resende, and G. Veiga An implementation of Karmarkar's algorithm for linear programming Math. Programming 44 1989 297 335
3. E.R. Barnes A variation on Karmarkar's algorithm for solving linear programming problems Math. Programming 36 1986 174 182
4. E.R. Barnes Some results concerning convergence of the affine scaling algorithm Contemp. Math. 114 1990 131 139
5. A. Bjorck Numerical Methods for Least Squares Problems 1996 SIAM Philadelphia
6. P. Bloomfield, and W.L. Steiger Least Absolute Deviations 1983 Birkhauser Boston
7. Cavalier, T.M., Soyster, A.L., 1985. Some computational experience and a modification of the Karmarkar algorithm. ISME Working Paper 85-105, The Pennsylvania State University.
8. Y. Censor, M.D. Altschuler, and W.D. Powlis A computational solution of the inverse problem in radiation-therapy treatment planning Appl. Math. Comput. 25 1988 57 87
9. Y. Censor, and T. Elfving New methods for linear inequalities Linear Algebra Appl. 42 1982 199 211
10. Y. Censor, and S.A. Zenios Parallel Optimization, Theory Algorithms, and Applications 1997 Oxford University Press Oxford
11. A. Dax The relationship between theorems of the alternative, least norm problems, steepest descent directions, and degeneracy a review Ann. Oper. Res. 46 1993 11 60
12. A. Dax Loss and retention of accuracy in affine scaling methods Optimi. Methods Software 15 2001 121 151
13. A. Dax The smallest correction of an inconsistent system of linear inequalities Optim. Eng. 2 2001 349 359
14. A. Dax Computing projections via Householder transformations and Gram-Schmidt orthogonalizations Numer. Linear Algebra Appl. 11 2004 675 692
15. A. Dax, and V.P. Sreedharan On theorems of the alternative and duality J. Optim. Theory Appl. 94 1997 561 590
16. A.R. De Pierro, and A.N. Iusem A simultaneous projection method for linear inequalities Linear Algebra Appl. 64 1985 243 253
17. I.I. Dikin Iterative solution of problems of linear and quadratic programming Sov. Math. Dokl. 8 1967 674 675
18. B. Efron Computer Intensive Methods in Statistical Regression SIAM Rev. 30 1988 421 449
19. P.E. Gill, W. Murray, and M.H. Wright Numerical Linear Algebra and Optimization 1991 Addison-Wesley Reading, MA
20. G.H. Golub, and C.F. Van Loan Matrix Computations 1983 Johns Hopkins University Press Baltimore
21. Han, S.P., 1980. Least-squares solution of linear inequalities. Technical Report 2141, Math. Res. Center, University of Wisconsin-Madison.
22. G.T. Herman A relaxation method for reconstructing object from noisy x-rays Math. Progr. 8 1975 1 19
23. G.T. Herman, and A. Lent A family of iterative quadratic optimization algorithms for pairs of inequalities, with applications in diagnostic radiology Math. Progr. Study 9 1978 15 29
24. A.N. Iusem, and A.R. De Pierro A simultaneous iterative method for computing projections on polyhedra SIAM J. Control 25 1987 231 243
25. N. Karmarkar A new polynomial-time algorithm for linear programming Combinatorica 4 1984 373 395
26. Leary, R.H., 1988. L 1 solution of overdetermined linear equations by an affine scaling linear programming algorithm. Technical Report, presented at the SIAM National Meeting, Minneapolis, 11 July.
27. A. Lent, and Y. Censor Extensions of Hildreth's row-action method for quadratic programming SIAM J. Control Optim. 18 1980 444 454
28. O.L. Mangasarian Nonlinear Programming 1969 McGraw-Hill New York
29. W.F. Mascarenhas The affine scaling algorithm fails for λ = 0.999 SIAM J. Optim. 7 1997 34 46
30. Meketon, M.S., 1987. Least absolute value regression. Technical Report, AT&T Bell Laboratory, Murray Hill, NJ.
31. Merz, P.H., 1998. An algorithm for percentile curves via constrained weighted L 1 regression, Technical Report, Chevron Research and Technology Company.
32. C.L. Monma, and A.J. Morton Computational experiments with a dual affine variant of Karmarkar's method for linear programming Oper. Res. Lett. 6 1987 261 267
33. R.D.C. Monteiro, T. Tsuchiya, and Y. Wang A simplified global convergence proof of the affine scaling algorithm Ann. Oper. Res. 47 1993 443 482
34. M. Muramatsu, and T. Tsuchiya A convergence analysis of the projective scaling algorithm based on the long-step homogeneous affine scaling algorithm Math. Programming 72 1996 291 305
35. J. Nocedal, and S. Wright Numerical Optimization 1999 Springer New York
36. S.A. Ruzinsky, and E.T. Olsen ℓ 1 and ℓ ∞ minimization via a variant of Karmarkar's algorithm IEEE Trans. Acoust. Speech Signal Process. 37 1989 245 253
37. Stewart, G.W., 1998. Matrix Algorithms, Volume 1: Basic Decompositions. SIAM, Philadelphia.
38. P. Tseng, and Z. Luo On the convergence of the affine-scaling algorithm Math. Programming 56 1992 301 319
39. T. Tsuchiya Global convergence of the affine scaling methods for degenerate linear programming problems Math. Programming Ser. B 52 1991 377 404
40. T. Tsuchiya Global convergence property of the affine scaling methods for primal degenerate linear programming problems Math. Oper. Res. 17 1992 527 557
41. T. Tsuchiya Affine scaling algorithm T. Terkaly Interior Point Methods of Mathematical Programming 1996 Kluwer Academic Publishers Dordrecht 35 82
42. T. Tsuchiya, and R.D.C. Monteiro Superlinear convergence of the affine scaling algorithm Math. Programming 75 1996 77 110
43. T. Tsuchiya, and M. Muramatsu Global convergence of a long-step affine scaling algorithm for degenerate linear programming problems SIAM J. Optim. 5 1995 525 551
44. R.J. Vanderbei Affine-scaling for linear programs with free variables Math. Programming 43 1989 31 44
45. R.J. Vanderbei, M.S. Meketon, and B.A. Freedman A modification of Karmarkar's linear programming algorithm Algorithmica 1 1986 395 407
46. G.A. Watson Choice of norms for data fitting and function approximation Acta Numer. 7 1998 337 377