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An entire space polynomial-time algorithm for linear programming

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  • Da Tian

Abstract

We propose an entire space polynomial-time algorithm for linear programming. First, we give a class of penalty functions on entire space for linear programming by which the dual of a linear program of standard form can be converted into an unconstrained optimization problem. The relevant properties on the unconstrained optimization problem such as the duality, the boundedness of the solution and the path-following lemma, etc, are proved. Second, a self-concordant function on entire space which can be used as penalty for linear programming is constructed. For this specific function, more results are obtained. In particular, we show that, by taking a parameter large enough, the optimal solution for the unconstrained optimization problem is located in the increasing interval of the self-concordant function, which ensures the feasibility of solutions. Then by means of the self-concordant penalty function on entire space, a path-following algorithm on entire space for linear programming is presented. The number of Newton steps of the algorithm is no more than $$O(nL\log (nL/ {\varepsilon }))$$ , and moreover, in short step, it is no more than $$O(\sqrt{n}\log (nL/{\varepsilon }))$$ . Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • Da Tian, 2014. "An entire space polynomial-time algorithm for linear programming," Journal of Global Optimization, Springer, vol. 58(1), pages 109-135, January.
    • Handle: RePEc:spr:jglopt:v:58:y:2014:i:1:p:109-135
    DOI: 10.1007/s10898-013-0048-z
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    References listed on IDEAS

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    1. E. A. Papa Quiroz & P. R. Oliveira, 2007. "New Self-Concordant Barrier for the Hypercube," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 475-490, December.
    2. O. Shevchenko, 2009. "Recursive Construction of Optimal Self-Concordant Barriers for Homogeneous Cones," Journal of Optimization Theory and Applications, Springer, vol. 140(2), pages 339-354, February.
    3. Frenk, J.B.G. & Gromicho, J.A.S. & Zhang, S., 1994. "A deep cut ellipsoid algorithm for convex programming," Econometric Institute Research Papers 11633, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. NESTEROV, Yu., 2006. "Constructing self-concordant barriers for convex cones," LIDAM Discussion Papers CORE 2006030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. R. Polyak & I. Griva, 2004. "Primal-Dual Nonlinear Rescaling Method for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 111-156, July.
    6. J. Burke & S. Xu, 2002. "Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 53-76, January.
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    Cited by:

    1. Da Tian, 2015. "An exterior point polynomial-time algorithm for convex quadratic programming," Computational Optimization and Applications, Springer, vol. 61(1), pages 51-78, May.

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