## A parameter-free penalty method for solving the linear programming problem |

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### Contents

PARAMETERFREE PENALTY METHOD | 12 |

PARAMETERFREE PENALTY METHOD USING BASIS MATRICES | 37 |

CONCLUDING REMARKS | 46 |

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active rows ak+1 APPENDIX arbitrary parameter au(x Bartels basic feasible point Chapter columns of Q compute Conn Conn's Consider the L.P. constraint matrix decrease descent direction dual element find a descent FORTRAN Hence I2V+ identity matrix independent columns index sets jth column KXXX KXXX L.P. problem line search Linear Algebra linear programming problem linearly independent m x m m x n matrix which projects minimize c x subject minimize p(x n x n n x t optimization problem overdetermined system parameter-free penalty function PARAMETER-FREE PENALTY METHOD phase problem T minimize projection matrix QK+1 QR decomposition QT Vp(x scheme sequence of Givens simplex algorithm Simplex method solution SOLVING THE LINEAR STEP storage sufficiently small Suppose a column TEMP2(K Theorem thesis transformations Update upper triangular form xxxx xxxx xxxxx xxxxx zero vector