## Progress in mathematical programming: interior-point and related methodsThe starting point of this volume was a conference entitled "Progress in Mathematical Programming," held at the Asilomar Conference Center in Pacific Grove, California, March 1-4, 1987. The main topic of the conference was developments in the theory and practice of linear programming since Karmarka's algorithm. Presentations included new algorithms, new analysis of algorithms, reports on computational experience, and some other topics related to the practice of mathematical programming. |

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### Contents

A PrimalDual Interior Point Algorithm for Linear Programming | 29 |

An Extension of Karmarkars Algorithm and the Trust Region | 49 |

Approximate Projections in a Projective Method for the Linear | 65 |

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### Common terms and phrases

algorithm for linear barrier function barrier function method barrier method bounded CG method complementary conceptual algorithm conjugate gradient consider convergence convex corresponding defined denote dual feasible dual linear programs dual problems given Hessian homotopy implementation inequality interior point interior point method Karmarkar's algorithm Karmarkar's method Karmarkar's projective Lemma linear complementarity problem linear equations linear programming problem logarithmic barrier function Math Mathematical Programming Megiddo method for linear N-R step Newton's method nonlinear programming null space null space affine number of iterations objective function objective value obtained optimal solution optimal value parameter path penalized function penalty multiplier polynomial polynomial-time algorithm polytope primal and dual projection matrix QR decomposition quadratic programming satisfies search direction Section sequence simplex algorithm simplex method solving space affine scaling subject to Ax system of linear techniques Theorem tion total number updates variables vector WHIZARD zero