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Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming

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  • Luo, Z-Q.
  • Sturm, J.F.
  • Zhang, S.

Abstract

This paper establishes the superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primal-dual optimal solution and that the size of the central path neighborhood tends to zero. The interior point algorithm considered here closely resembles the Mizuno-Todd-Ye predictor-corrector method for linear programming which is known to be quadratically convergent. It is shown that when the iterates are well centered, the duality gap is reduced superlinearly after each predictor step. Indeed, if each predictor step is succeeded by [TeX: $r$] consecutive corrector steps then the predictor reduces the duality gap superlinearly with order [TeX: $\\frac{2}{1+2^{-2r}}$]. The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primal-dual central path converges to the analytic center of the primal-dual optimal solution set, and the distance from any point on the central path to this analytic center is bounded by the duality gap.

Suggested Citation

  • Luo, Z-Q. & Sturm, J.F. & Zhang, S., 1996. "Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming," Econometric Institute Research Papers 9607/A, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:1373
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    References listed on IDEAS

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    1. Shinji Mizuno & Michael J. Todd & Yinyu Ye, 1993. "On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 964-981, November.
    2. NESTEROV , Yurii & TODD , Michael, 1995. "Primal-Dual Interior-Point Methods for Self-Scaled Cones," LIDAM Discussion Papers CORE 1995044, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. R. D. C. Monteiro & Jong-Shi Pang, 1998. "On Two Interior-Point Mappings for Nonlinear Semidefinite Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 39-60, February.
    2. 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.
    3. J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.

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