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Andre L. Tits (andreumd.edu) Abstract: Consider solving a linear program in standard form, where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semiinfinite program. The cost per iteration of typical primaldual interiorpoint methods on such problems is $O(m^2n)$. We propose to reduce that cost by replacing the normal equation matrix, $AD^2A^{\T}$, $D$ a diagonal matrix, with a ``reduced'' version (of same dimension), $A_QD_Q^2A_Q^{\T}$, where $Q$ is an index set including the indices of $M$ most nearly active (or most violated) dual constraints at the current iterate, $M\geq m$ a prescribed integer. This can result in a speedup of close to $n/Q$ at each iteration. Promising numerical results are reported for constraintreduced versions of a dualfeasible affinescaling algorithm and of Mehrotra's PredictorCorrector method [SIAM J. Opt., Vol.2, pp. 575601, 1992]. In particular, while it could be expected that neglecting a large portion of the constraints, especially at early iterations, may result in a significant deterioration of the search direction, it turns out that the total number of iterations remains essentially constant as the size of the reduced constraint set is decreased, often down to a small fraction of the total set. In the case of the affinescaling algorithm, global convergence and local quadratic convergence are proved. Keywords: linear programming, constraint reduction, column generation, primaldual interiorpoint methods, affine scaling, Mehrotra's PredictorCorrector Category 1: Linear, Cone and Semidefinite Programming (Linear Programming ) Citation: SIAM J.on Optimization, Vol.17, No.1, pp.119146, 2006. Download: Entry Submitted: 04/28/2005 Modify/Update this entry  
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