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Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory

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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 the solutions of the system of ODEs. In Sim and Zhao (Math. Program. Ser. A, [2007], to appear), 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). Off-central paths for a simple example are also studied in Sim and Zhao (Math. Program. Ser. A, [2007], to appear) and their asymptotic behavior near the solution of the example is analyzed. In this paper, which is an extension of Sim and Zhao (Math. Program. Ser. A, [2007], to appear), we study the asymptotic behavior of the off-central paths for general SDLCPs using the dual HKM direction. We give a necessary and sufficient condition for when an off-central path is analytic as a function of \(\sqrt{\mu}\) at a solution of the SDLCP. Then, we show that, if the given SDLCP has a unique solution, the first derivative of its off-central path, as a function of \(\sqrt{\mu}\) , is bounded. We work under the assumption that the given SDLCP satisfies the strict complementarity condition.

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Authors and Affiliations

  1. Department of Decision Sciences, National University of Singapore Business School, Singapore, Singapore

    C. K. Sim

  2. Department of Mathematics, National University of Singapore, Singapore, Singapore

    G. Zhao

Authors
  1. C. K. Sim
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  2. G. Zhao
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Correspondence to C. K. Sim.

Additional information

Communicated by P. Tseng.

This research was done during the first author’s Ph.D. study at the Department of Mathematics, National University of Singapore, and as a Research Engineer at the NUS Business School. The second author’s work was supported in part by NUS Academic Research Grant R-146-000-084-112.

For technical details of the paper, please refer to The Logistics Institute—Asia Pacific Internal Report. See also http://www.math.nus.edu.sg/~matzgy/publist.html.

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Sim, C.K., Zhao, G. Asymptotic Behavior of Helmberg-Kojima-Monteiro (HKM) Paths in Interior-Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory. J Optim Theory Appl 137, 11–25 (2008). https://doi.org/10.1007/s10957-007-9280-3

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