Abstract
We propose a new framework for the application of preconditioned conjugate gradients in the solution of large-scale linear equality constrained minimization problems. This framework allows for the exploitation of structure and sparsity in the context of solving the reduced Newton system (despite the fact that the reduced system may be dense). Numerical experiments performed on a variety of test problems from the Netlib LP collection indicate computational promise.
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References
L. Adams and J. Nazareth, “Linear and nonlinear conjugate gradient-related methods,” SIAM, 1996.
T.F. Coleman, “Linearly constrained optimization and projected preconditioned conjugate gradients,” in Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, SIAM: Philadelphia, 1994, pp. 118–122.
T.F. Coleman and A. Pothen, “he null space problem I: Complexity,” SIAM J. Alg. & Disc. Meth., vol. 7, pp. 527–537, 1987.
T.F. Coleman and A. Pothen, “The null space problem II: Algorithms,” SIAM J. Alg. & Disc. Meth., vol. 8, pp. 544–563, 1987.
T.F. Coleman and A. Verma, “Structure and efficient Jacobian calculation,” in Computational Differentiation: Techniques, Applications, and Tools, M. Berz, C. Bischof, G. Corliss, and A. Griewank (Eds.), SIAM: Philadelphia, PA, pp. 149–159, 1996.
T.F. Coleman and A. Verma, “Structure and efficient Hessian calculation,” in Advances in Nonlinear Programming, Proceedings of the 1996 International Conference on Nonlinear Programming, Y.-X. Yuan (Ed.), Kluwer Academic Publishers: Dordrecht, The Netherlands, pp. 57–72, 1998.
J.R. Gilbert and M. Heath, “Computing a sparse basis for the nullspace,” SIAM J. Alg. & Disc. Meth., vol.8, pp.446–459, 1987.
P. Gill, W. Murray, D. Ponceleon, and M. Saunders, “Preconditioners for indefinite systems arising in optimization,” SIAM J. Matrix Anal. Appl., vol. 13, pp. 292–311, 1992.
J.J. Moré, B.S. Garbow, and K.H. Hillstrom, “Testing unconstrained optimization software,” ACM Trans. on Math. Software, no. 7, pp. 17–41, 1981.
S.G. Nash and A. Sofer, “Preconditioning of reduced matrices,” SIAM Journal on Matrix Analysis, vol. 17, pp. 47–68, 1996.
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Coleman, T.F., Verma, A. A Preconditioned Conjugate Gradient Approach to Linear Equality Constrained Minimization. Computational Optimization and Applications 20, 61–72 (2001). https://doi.org/10.1023/A:1011271406353
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DOI: https://doi.org/10.1023/A:1011271406353