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On Sensitivity of Central Solutions in Semidefinite Programming

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  • J.F. Sturm

    (McMaster University, Hamilton, Canada)

  • S. Zhang

    (Erasmus University Rotterdam)

Abstract

In this paper we study the properties of the analytic central path of asemidefinite programming problem under perturbation of a set of inputparameters. Specifically, we analyze the behavior of solutions on the centralpath with respect to changes on the right hand side of the constraints,including the limiting behavior when the central optimal solution isapproached. Our results are of interest for the sake of numerical analysis,sensitivity analysis and parametric programming.Under the primal-dual Slater condition and the strict complementarity conditionwe show that the derivatives of central solutions with respect to theright hand side parameters converge as the path tends to the centraloptimal solution. Moreover, the derivatives are bounded, i.e. aLipschitz constant exists.This Lipschitz constant can be thought of as a condition number for thesemidefinite programming problem. It is a generalization of the familiarcondition number for linear equation systems and linear programming problems.However, the generalized condition number depends on the right hand sideparameters as well, whereas it is well-known that in the linear programming casethe condition number depends only on the constraint matrix.We demonstrate that the existence of strictly complementary solutionsis important for the Lipschitz constant to exist.Moreover, we give an example in which the set of right hand side parameters forwhich the strict complementarity condition holds is neither open nor closed.This is remarkable since a similar set for which the primal-dual Slatercondition holds is always open.

Suggested Citation

  • J.F. Sturm & S. Zhang, 1998. "On Sensitivity of Central Solutions in Semidefinite Programming," Tinbergen Institute Discussion Papers 98-040/4, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:19980040
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    References listed on IDEAS

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    1. Michael J. Todd, 1990. "A Dantzig-Wolfe-Like Variant of Karmarkar's Interior-Point Linear Programming Algorithm," Operations Research, INFORMS, vol. 38(6), pages 1006-1018, December.
    2. 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.
    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. 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.
    5. Nunez, M. A. (Manuel A.) & Freund, Robert Michael. & Massachusetts Institute of Technology. Operations Research Center., 1996. "Condition measures and properties of the central trajectory of a linear program," Working papers 316-96., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    6. 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).
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    Cited by:

    1. E. A. Yıldırım, 2003. "An Interior-Point Perspective on Sensitivity Analysis in Semidefinite Programming," Mathematics of Operations Research, INFORMS, vol. 28(4), pages 649-676, November.
    2. Yoshiyuki Sekiguchi & Hayato Waki, 2021. "Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 52-72, January.
    3. Zhang, S., 1998. "Global error bounds for convex conic problems," Econometric Institute Research Papers EI 9830, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.

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