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Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming

Author

Listed:
  • NESTEROV ., Yurii E.

    (CORE, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium)

  • TODD , Michael J

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

Abstract

This paper provides a theoretical foundation for efficient interior-point algorithms for nonlinear programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

Suggested Citation

  • NESTEROV ., Yurii E. & TODD , Michael J, 1994. "Self-Scaled Cones and Interior-Point Methods in Nonlinear Programming," LIDAM Discussion Papers CORE 1994062, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1994062
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1994.html
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    Citations

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    Cited by:

    1. Berkelaar, A.B. & Sturm, J.F. & Zhang, S., 1996. "Polynomial Primal-Dual Cone Affine Scaling for Semidefinite Programming," Econometric Institute Research Papers EI 9667-/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    2. Terlaky, Tamas, 2001. "An easy way to teach interior-point methods," European Journal of Operational Research, Elsevier, vol. 130(1), pages 1-19, April.
    3. Helmberg, C., 2002. "Semidefinite programming," European Journal of Operational Research, Elsevier, vol. 137(3), pages 461-482, March.
    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. Ali Mohammad-Nezhad & Tamás Terlaky, 2017. "A polynomial primal-dual affine scaling algorithm for symmetric conic optimization," Computational Optimization and Applications, Springer, vol. 66(3), pages 577-600, April.
    6. Maziar Salahi & Renata Sotirov & Tamás Terlaky, 2004. "On self-regular IPMs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 12(2), pages 209-275, December.
    7. Sturm, J.F., 2001. "Avoiding Numerical Cancellation in the Interior Point Method for Solving Semidefinite Programs," Discussion Paper 2001-27, Tilburg University, Center for Economic Research.
    8. Arjan B. Berkelaar & Jos F. Sturm & Shuzhong Zhang, 1997. "Polynomial Primal-Dual Cone Affine Scaling for Semidefinite Programming," Tinbergen Institute Discussion Papers 97-025/4, Tinbergen Institute.
    9. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    10. Sturm, J.F., 2002. "Implementation of Interior Point Methods for Mixed Semidefinite and Second Order Cone Optimization Problems," Discussion Paper 2002-73, Tilburg University, Center for Economic Research.
    11. Alemseged Weldeyesus & Mathias Stolpe, 2015. "A primal-dual interior point method for large-scale free material optimization," Computational Optimization and Applications, Springer, vol. 61(2), pages 409-435, June.
    12. NESTEROV, Yu., 2006. "Towards nonsymmetric conic optimization," LIDAM Discussion Papers CORE 2006028, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

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