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Primal-Dual Methods and Infeasibility Detectors for Nonlinear Programming Problems

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  • NESTEROV, Yurii

    (Center for Operations Research and Econometrics (CORE), Université catholique de Louvain (UCL), Louvain la Neuve, Belgium)

  • TODD, Michael

    (School of Operations Research and Industrial Engineering, Cornell University, Ithaca)

  • YE, Yinyu

    (The University of Iowa, Iowa City)

Abstract

In this paper we present several "infeasible-start" path-following and potential-reduction primal-dual interior-point methods for nonlinear conic problems. These methods try to find a recession direction of the feasible set of a self-dual homogeneous primal-dual problem. The methods under consideration generate an E -solution for an E- perturbation of an initial strictly (primal and dual) feasible problem in O [square root. v ln(v /e pf)] iterations, where v is the parameter of a self-concordant barrier for the cone, E is a relative accuracy and pf is a feasibility measure. We also discuss the behavior of path-following methods as applied to infeasible problems. We prove that strict infeasibility (primal or dual) can be detected in O [square root. v ln(v /p)] iterations, where p. is a primal or dual infeasibility measure.

Suggested Citation

  • NESTEROV, Yurii & TODD, Michael & YE, Yinyu, 1996. "Primal-Dual Methods and Infeasibility Detectors for Nonlinear Programming Problems," LIDAM Discussion Papers CORE 1996037, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1996037
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1996.html
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    Cited by:

    1. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1997. "Duality Results for Conic Convex Programming," Econometric Institute Research Papers EI 9719/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1998. "Conic convex programming and self-dual embedding," Econometric Institute Research Papers EI 9815, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    3. Levent Tunçel, 1998. "Primal-Dual Symmetry and Scale Invariance of Interior-Point Algorithms for Convex Optimization," Mathematics of Operations Research, INFORMS, vol. 23(3), pages 708-718, August.
    4. Liqun Qi & Yinyu Ye, 2014. "Space tensor conic programming," Computational Optimization and Applications, Springer, vol. 59(1), pages 307-319, October.
    5. Chris Coey & Lea Kapelevich & Juan Pablo Vielma, 2022. "Solving Natural Conic Formulations with Hypatia.jl," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2686-2699, September.

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