A fixed matrix iteration algorithm was proposed by A. Dax (Numer. Algor. 50, 97–114 2009) for solving linear inequalities in a least squares sense. However, a great deal of computation for this algorithm is required, especially for large-scale problems, because a least squares subproblem should be solved accurately at each iteration. We present a modified method, the inexact fixed iteration method, which is a generalization of the fixed matrix iteration method. In this inexact iteration process, the classical LSQR method is implemented to determine an approximate solution of each least squares subproblem with less computational effort. The convergence of this algorithm is analyzed and several numerical examples are presented to illustrate the efficiency of the inexact fixed matrix iteration algorithm for solving large linear inequalities.
Linear inequalities Inconsistent systems Fixed matrix iteration Inexact fixed matrix iteration Krylov subspace LSQR method
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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)CrossRefMATHMathSciNetGoogle Scholar
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)CrossRefMATHMathSciNetGoogle Scholar
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)Google Scholar
Golub, G.H., Kahan, W.: Calculating the singular values and pseudoinverse of a matrix. SIAM J. Numer. Anal 2, 205–224 (1965)MATHMathSciNetGoogle Scholar
Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press (1983)Google Scholar
Han, S.P.: Least squares solution of linear inequalities. Technical Report 2141, Math. Res. Center, University of Wisconsin-Madison (1980)Google Scholar
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)CrossRefMATHMathSciNetGoogle Scholar
Stiefel, E.: Ausgleichung ohne Aufstellung der Gausschen Normalgleichungen. Wiss. Z. Tech. Hochsch. Dresden 2, 441–442 (1952–1953)MathSciNetGoogle Scholar
Wang, R.S.: Functional Analysis and Optimization Theory. Beijing University of Aeronautics and Astronautics Press (2003). In ChineseGoogle Scholar
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)CrossRefMATHMathSciNetGoogle Scholar
Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim 43, 147–168 (2001)CrossRefMATHMathSciNetGoogle Scholar
Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press (1971)Google Scholar