Gould, Nicholas I. M.; Hribar, Mary E.; Nocedal, Jorge On the solution of equality constrained quadratic programming problems arising in optimization. (English) Zbl 0999.65050 SIAM J. Sci. Comput. 23, No. 4, 1376-1395 (2001). Summary: 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. Cited in 79 Documents MSC: 65K05 Numerical mathematical programming methods 90C55 Methods of successive quadratic programming type 65F35 Numerical computation of matrix norms, conditioning, scaling 90C06 Large-scale problems in mathematical programming 90C30 Nonlinear programming Keywords:nonlinear optimization; conjugate gradient method; quadratic programming; preconditioning; iterative refinement; numerical examples Software:CUTEr; TRICE; mctoolbox PDFBibTeX XMLCite \textit{N. I. M. Gould} et al., SIAM J. Sci. Comput. 23, No. 4, 1376--1395 (2001; Zbl 0999.65050) Full Text: DOI