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Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems

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  • Chee-Khian Sim

    (The Hong Kong Polytechnic University)

Abstract

An interior point method (IPM) defines a search direction at each interior point of the feasible region. These search directions form a direction field, which in turn gives rise to a system of ordinary differential equations (ODEs). Thus, it is natural to define the underlying paths of the IPM as solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), these off-central paths are shown to be well-defined analytic curves and any of their accumulation points is a solution to the given monotone semidefinite linear complementarity problem (SDLCP). In Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007; J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of off-central paths corresponding to the HKM direction is studied. In particular, in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), the authors study the asymptotic behavior of these paths for a simple example, while, in Sim and Zhao (J. Optim. Theory Appl. 137:11–25, 2008) and Sim (J. Optim. Theory Appl. 141:193–215, 2009), the asymptotic behavior of these paths for a general SDLCP is studied. In this paper, we study off-central paths corresponding to another well-known direction, the Nesterov-Todd (NT) direction. Again, we give necessary and sufficient conditions for these off-central paths to be analytic w.r.t. $\sqrt{\mu}$ and then w.r.t. μ, at solutions of a general SDLCP. Also, as in Sim and Zhao (Math. Program. Ser. A 110:475–499, 2007), we present off-central path examples using the same SDP, whose first derivatives are likely to be unbounded as they approach the solution of the SDP. We work under the assumption that the given SDLCP satisfies a strict complementarity condition.

Suggested Citation

  • Chee-Khian Sim, 2011. "Asymptotic Behavior of Underlying NT Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 79-106, January.
    • Handle: RePEc:spr:joptap:v:148:y:2011:i:1:d:10.1007_s10957-010-9746-6
    DOI: 10.1007/s10957-010-9746-6
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    References listed on IDEAS

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    1. F. A. Potra & R. Sheng, 1998. "Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 103-119, October.
    2. Jos F. Sturm, 1999. "Superlinear Convergence of an Algorithm for Monotone Linear Complementarity Problems, When No Strictly Complementary Solution Exists," Mathematics of Operations Research, INFORMS, vol. 24(1), pages 72-94, February.
    3. Yu. E. Nesterov & M. J. Todd, 1997. "Self-Scaled Barriers and Interior-Point Methods for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 1-42, February.
    4. Sanjay Mehrotra, 1993. "Quadratic Convergence in a Primal-Dual Method," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 741-751, August.
    5. C. K. Sim, 2009. "On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 193-215, April.
    6. C. K. Sim & G. Zhao, 2008. "Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory," Journal of Optimization Theory and Applications, Springer, vol. 137(1), pages 11-25, April.
    7. F. A. Potra & R. Sheng, 1998. "Superlinearly Convergent Infeasible-Interior-Point Algorithm for Degenerate LCP," Journal of Optimization Theory and Applications, Springer, vol. 97(2), pages 249-269, May.
    8. Josef Stoer & Martin Wechs & Shinji Mizuno, 1998. "High Order Infeasible-Interior-Point Methods for Solving Sufficient Linear Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(4), pages 832-862, November.
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    Cited by:

    1. Chee-Khian Sim, 2019. "Interior point method on semi-definite linear complementarity problems using the Nesterov–Todd (NT) search direction: polynomial complexity and local convergence," Computational Optimization and Applications, Springer, vol. 74(2), pages 583-621, November.

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