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 Karmarkar's algorithm. There were thirty presentations and approximately fifty people attended. Presentations included new algorithms, new analyses of algorithms, reports on computational experience, and some other topics related to the practice of mathematical programming. Interestingly, most of the progress reported at the conference was on the theoretical side. Several new polynomial algorithms for linear program ming were presented (Barnes-Chopra-Jensen, Goldfarb-Mehrotra, Gonzaga, Kojima-Mizuno-Yoshise, Renegar, Todd, Vaidya, and Ye). Other algorithms presented were by Betke-Gritzmann, Blum, Gill-Murray-Saunders-Wright, Nazareth, Vial, and Zikan-Cottle. Efforts in the theoretical analysis of algo rithms were also reported (Anstreicher, Bayer-Lagarias, Imai, Lagarias, Megiddo-Shub, Lagarias, Smale, and Vanderbei). Computational experiences were reported by Lustig, Tomlin, Todd, Tone, Ye, and Zikan-Cottle. Of special interest, although not in the main direction discussed at the conference, was the report by Rinaldi on the practical solution of some large traveling salesman problems. At the time of the conference, it was still not clear whether the new algorithms developed since Karmarkar's algorithm would replace the simplex method in practice. Alan Hoffman presented results on conditions under which linear programming problems can be solved by greedy algorithms. |
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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Other editions - View all
Progress in Mathematical Programming: Interior-Point and Related Methods Nimrod Megiddo Limited preview - 2012 |
Progress in Mathematical Programming: Interior-Point and Related Methods Nimrod Megiddo No preview available - 2011 |
Common terms and phrases
algorithm for linear barrier function barrier function method barrier method bounded CG method complementary conjugate gradient consider convergence convex corresponding defined denote dual feasible dual linear programs dual problems Hessian homotopy implementation inequality interior point Karmarkar's algorithm Karmarkar's method Karmarkar's projective Lemma linear complementarity problem linear equations linear programming problem logarithmic barrier function Math Mathematical Programming Maximize Megiddo method for linear minimize N-R step Nazareth Newton's method nonlinear programming null space null space affine number of iterations objective function objective value obtained optimal solution parameter path 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 total number updates variables vector WHIZARD xk+1 zero