×

The inexact fixed matrix iteration for solving large linear inequalities in a least squares sense. (English) Zbl 1322.65068

The author presents a modified method, the inexact fixed iteration method for solving large linear inequalities, which is a generalization of the fixed matrix iteration method.

MSC:

65K05 Numerical mathematical programming methods
90C05 Linear programming
Full Text: DOI

References:

[1] Björck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996) · Zbl 0847.65023 · doi:10.1137/1.9781611971484
[2] Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press (2004) · Zbl 1058.90049
[3] Bramley, R., Winnicka, B.: Solving linear inequalities in a least squares sense. SIAM J. Sci. Comput 17, 275-286 (1996) · Zbl 0844.65024 · doi:10.1137/0917020
[4] Censor, Y., Altschuler, M.D., Powlis, W.D.: A computational solution of the inverse problem in radiation-therapy treatment planning. Appl. Math. Comput 25, 57-87 (1988) · Zbl 0637.65060 · doi:10.1016/0096-3003(88)90064-1
[5] Censor, Y., Ben-Israel, A., Xiao, Y., Galvin, J.M.: On linear infeasibility arising in intensity modulated radiation therapy inverse planning. Linear Algebra Appl 428, 1406-1420 (2008) · Zbl 1130.92030 · doi:10.1016/j.laa.2007.11.001
[6] Chinneck, J.W.: Feasibility and infeasibility in optimization: algorithms and computational methods. In: International Series in Operations Research and Management Sciences, vol. 118. Springer-Verlag (2007) · Zbl 1178.90369
[7] Davis, T.A.: Algorithm 915, SuiteSparseQR: Multifrontal multithreaded rank-revealing sparse QR factorization. ACM Trans. Math. Softw. 38, 1-22 (2011) · Zbl 1365.65122
[8] Dax, A.: The convergence of linear stationary iterative processes for solving singular unstructured systems of linear equations. SIAM Rev. 32, 611-635 (1990) · Zbl 0718.65021 · doi:10.1137/1032122
[9] Dax, A.: The smallest correction of an inconsistent system of linear inequalities. Optimiz. Eng. 2, 349-359 (2001) · Zbl 1050.90538 · doi:10.1023/A:1015370617219
[10] Dax, A.: A hybrid algorithm for solving linear inequalities in a least squares sense. Numer. Algor. 50, 97-114 (2009) · Zbl 1165.65029 · doi:10.1007/s11075-008-9218-3
[11] Gilbert, J.R., Moler, C., Schreiber, R.: Sparse matrices in MATLAB: design and implementation. SIAM J. Matrix Anal. Appl 13, 333-356 (1992) · Zbl 0752.65037 · doi:10.1137/0613024
[12] Golub, G.H., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal 2, 205-224 (1965) · Zbl 0194.18201
[13] Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press (1983) · Zbl 0559.65011
[14] Han, S.P.: Least squares solution of linear inequalities. Technical Report 2141, Math. Res. Center, University of Wisconsin-Madison (1980) · Zbl 0718.65021
[15] He, B.S.: A projection and contraction method for a class of linear complementarity problem and its application in convex quadratic programming. Appl. Math. Optim. 25, 247-262 (1992) · Zbl 0767.90086 · doi:10.1007/BF01182323
[16] He, B.S.: Solving a class of linear projection equations. Numer. Math. 68, 71-80 (1994) · Zbl 0822.65040 · doi:10.1007/s002110050048
[17] Herman, G.T.: A relaxation method for reconstructing object from noisy X-rays. Math. Progr. 8, 1-19 (1975) · Zbl 0298.65028 · doi:10.1007/BF01580425
[18] Herman, G.T.: Image Reconstruction from Projections: The Fundamentals of Computerized Tomography. Academic Press, New York (1982) · Zbl 0538.92005
[19] Moreau, J.J.: Decomposition orthogonale d’un espace Hilbertien selon deux cânes mutuellement polaires. C.R. Acad. Sci. Paris 225, 238-240 (1962) · Zbl 0109.08105
[20] Morikuni, K., Hayami, K.: Inner-iteration Krylov subspace methods for least squares problems. SIAM J. Matrix Anal. Appl 34, 1-22 (2013) · Zbl 1269.65039 · doi:10.1137/110828472
[21] Paige, C.C., Saunders, M.A.: LSQR, An algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8, 43-71 (1982) · Zbl 0478.65016 · doi:10.1145/355984.355989
[22] Pinar, M.C.: Newton’s method for linear inequality systems. Eur. J. Oper. Res 107, 710-719 (1998) · Zbl 0927.90099 · doi:10.1016/S0377-2217(97)00178-1
[23] Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003) · Zbl 1031.65046 · doi:10.1137/1.9780898718003
[24] Stewart, G.W.: Matrix Algorithms, Volume 1: Basic Decompositions. SIAM, Philadelphia (1998) · Zbl 0910.65012 · doi:10.1137/1.9781611971408
[25] Stiefel, E.: Ausgleichung ohne Aufstellung der Gausschen Normalgleichungen. Wiss. Z. Tech. Hochsch. Dresden 2, 441-442 (1952-1953) · Zbl 0051.34901
[26] Wang, R.S.: Functional Analysis and Optimization Theory. Beijing University of Aeronautics and Astronautics Press (2003). In Chinese
[27] Wierzbicki, A.P., Kurcyusz, S.: Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space. SIAM J. Control Optim 15, 25-56 (1977) · Zbl 0355.90045 · doi:10.1137/0315003
[28] Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim 43, 147-168 (2001) · Zbl 0980.90093 · doi:10.1007/s002450010023
[29] Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press (1971) · Zbl 0281.47043
[30] National Institute of Standards and Technology, MatrixMarket, http://math.nist.gov/MatrixMarket (2002) · Zbl 0927.90099
[31] Sparse QR factorization package, http://www.cise.ufl.edu/research/sparse/SPQR/ (2013) · Zbl 0822.65040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.