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On the Interplay among Entropy, Variable Metrics and Potential Functions in Interior-Point Algorithms

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Abstract

We are motivated by the problem of constructing aprimal-dual barrier function whose Hessian induces the (theoreticallyand practically) popular symmetric primal and dual scalings forlinear programming problems. Although this goal is impossible toattain, we show that the primal-dual entropy function may provide asatisfactory alternative. We study primal-dual interior-pointalgorithms whose search directions are obtained from a potentialfunction based on this primal-dual entropy barrier. We providepolynomial iteration bounds for these interior-point algorithms. Thenwe illustrate the connections between the barrier function and areparametrization of the central path equations. Finally, we considerthe possible effects of more general reparametrizations oninfeasible-interior-point algorithms.

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Authors and Affiliations

  1. Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Levent Tuncel

  2. School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY, 14853-3801, U.S.A

    Michael J. Todd

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  1. Levent Tuncel
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  2. Michael J. Todd
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Tuncel, L., Todd, M.J. On the Interplay among Entropy, Variable Metrics and Potential Functions in Interior-Point Algorithms. Computational Optimization and Applications 8, 5–19 (1997). https://doi.org/10.1023/A:1008637929927

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