"On the solution of equality constrained quadratic programming problems arising in optimization "
N.I.M. Gould, M.E. Hribar, J. Nocedal.
SIAM J. Sci. Computing, Vol 23, No.4, pp. 1375-1394 (2001)

We consider the application of the conjugate gradient method to the solution of large equality constrained quadratic programs arising in nonlinear optimization. Our approach is based implicitly on a reduced linear system and generates iterates in the null space of the constraints. Instead of computing a basis for this null space, we choose to work directly with the matrix of constraint gradients, computing projections into the null space by either a normal equations or an augmented system approach. Unfortunately, in practice such projections can result in significant rounding errors. We propose iterative refinement techniques, as well as an adaptive reformulation of the quadratic problem, that can greatly reduce these errors without incurring high computational overheads. Numerical results illustrating the efficacy of the proposed approaches are presented.
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