作者:
GONZAGA, CC1. COPPE
Federal University of Rio de Janeiro P.O. Box 68511 21945 Rio de Janeiro RJ Brazil
Interior methods for linear programming were designed mainly for problems formulated with equality constraints and non-negative variables. The formulation with inequality constraints has shown to be very convenient fo...
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Interior methods for linear programming were designed mainly for problems formulated with equality constraints and non-negative variables. The formulation with inequality constraints has shown to be very convenient for practical implementations, and the translation of methods designed for one formulation into the other is not trivial. This paper relates the geometric features of both representations, shows how to transport data and procedures between them and shows how cones and conical projections can be associated with inequality constraints.
We devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution. Inclusion of such a constraint allows the algorithm to be applied to a (possibly infeasib...
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We devise a projective algorithm which explicitly considers the constraint that an artificial variable be zero at the solution. Inclusion of such a constraint allows the algorithm to be applied to a (possibly infeasible) standard form linear program, without the addition of any “bigM“ terms or conversion to a primal-dual problem.
In his original analysis of the projective algorithm for linear programming, Karmarkar proposed a “modified method” which improved the worst-case arithmetic complexity of the original algorithm by a factor of $\sqrt...
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In his original analysis of the projective algorithm for linear programming, Karmarkar proposed a “modified method” which improved the worst-case arithmetic complexity of the original algorithm by a factor of $\sqrt n $ . Karmarkar's analysis of the $\sqrt n $ improvement is based on a primal-dual formulation, and requires that small steps be taken on each iteration. However, in practice the original algorithm can be easily applied to standard form problems, and is considerably improved by performing a linesearch on each iteration, generally leading to much larger steps. We show here that by incorporating a simple safeguard, a linesearch may be performed in the modified algorithm while retaining the $\sqrt n $ complexity improvement over the original algorithm. We then show that the modified algorithm, with safeguarded linesearch, can be applied directly to a standard form linear program with unknown optimal objective value. The resulting algorithm enjoys a $\sqrt n $ complexity advantage over standard form variants of the original algorithm.
We demonstrate that Karmarkar's projective algorithm is fundamentally an algorithm for fractional linear programming on the simplex. Convergence for the latter problem is established assuming only an initial lower...
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We demonstrate that Karmarkar's projective algorithm is fundamentally an algorithm for fractional linear programming on the simplex. Convergence for the latter problem is established assuming only an initial lower bound on the optimal objective value. We also show that the algorithm can be easily modified so as to assure monotonicity of the true objective values, while retaining all global convergence properties. Finally, we show how the monotonic algorithm can be used to obtain an initial lower bound when none is otherwise available.
The most time-consuming part of the Karmarkar algorithm for linear programming is the projection of a vector onto the nullspace of a matrix that changes at each iteration. We present a variant of the Karmarkar algorit...
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The most time-consuming part of the Karmarkar algorithm for linear programming is the projection of a vector onto the nullspace of a matrix that changes at each iteration. We present a variant of the Karmarkar algorithm that uses standard variable-metric techniques in an innovative way to approximate this projection. In limited tests, this modification greatly reduces the number of matrix factorizations needed for the solution of linear programming problems.
As described in Goldfarb and Mehrotra [3], the convergence analysis of Karmarkar's method for linear programming leads naturally to an acceptance criterion when approximate projections are used in the course of th...
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As described in Goldfarb and Mehrotra [3], the convergence analysis of Karmarkar's method for linear programming leads naturally to an acceptance criterion when approximate projections are used in the course of the algorithm. Unfortunately, the preliminary results of Goldfarb and Mehrotra indicate that their criterion is generally not strong enough to insure that the algorithm makes reasonable progress. In this note we present a stronger criterion and show that the difference between the two criteria is potentially large.
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