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Spectral properties of matrices arising from primal-dual interior-point methods for convex quadratic programs

University of British Columbia

The solution of convex quadratic programs using primal-dual interior-point methods has at its core the solution of a series of linear systems, which are in practice commonly reduced by block Gaussian elimination from the original unsymmetric block 3-by-3 formulation to either a block 2-by-2 saddle-point matrix or a block 1-by-1 normal equations form. The 3-by-3 formulation can also be symmetrized with a similarity transformation if a symmetric solver is to be used. We examine whether this practice of reduction is beneficial for the solver. For each of these formulations we find analytical results for the following spectral properties: the inertia, condition number, conditions for nonsingularity, and bounds on the eigenvalues. While the reduced systems become increasingly ill-conditioned throughout the iterations except in special cases, the 3-by-3 formulations remain nonsingular and well-conditioned with only mild assumptions on the problem; with regularization the assumptions are further simplified. Numerical examples are used to support the analytical results. We conclude that the 3-by-3 formulations, unsymmetric or symmetric, unregularized or regularized, have superior spectral properties that support their use in practice.

Text

2012-08-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/43088

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/