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A convex optimization approach for solving large scale linear systems
Debora Cores, Johanna Figueroa

The well-known  Conjugate Gradient (CG) method minimizes a strictly convex quadratic function  for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the  characteristics of the coefficient matrix.  For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG) method, which allow us to extend this optimization approach for solving  consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies.  Our numerical results indicate that the new scheme outperforms  state-of-the-art  iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear
feasibility problems.

Keywords: Nonlinear convex optimization, spectral gradient method, large-scale linear systems

Cite this paper:
Cores D., Figueroa J., A convex optimization approach for solving large scale linear systems.
Bull. Comput. Appl. Math. (Bull CompAMa),
Vol. 5, No. 1, Jan-Jun, pp.53-76, 2017