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pti n algorit ically require the solution of many systems of linear equations S-l la-a nmbeconstraints are present, these linear systm cod the total computation time. Both direct d iterativ ion solvers are needed in ractice. Unfortunately, most of the off-the shelf solver e designed fo4ngle systems, whereas ptimization problems give rise to hundreds or thousan Of Stems. To id refactorization, r to speed the convergence of an 'ertive method, it is e tial to note thatj is related to -1. t .vsJW review various spars that arise in optimization, and discuss compromises that a Z i're currently being made in dealing with . Since significant advances continue to be made with single-system solverf#J give special atten to methods that allow such solvers to be used repeatedly on a sequence of modified systems (e.g., he product-form update; use of the Schur complement). The speed of factorizing a matrix then omes relatively less important than the efficiency of subsequent 10 w5 very many right-han sides. At the same time, we lthat future improvemen s to linear-equation software will be oriented more specifically to the cue of related matrices h'

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