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Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem

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Abstract

An interior point method defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves off-central paths. We study off-central paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a well-defined analytic curve with parameter μ ranging over (0, ∞) and any accumulation point of the off-central path is a solution to SDLCP. Through a simple example we show that the off-central paths are not analytic as a function of \(\sqrt{\mu}\) and have first derivatives which are unbounded as a function of μ at μ  =  0 in general. On the other hand, for the same example, we can find a subset of off-central paths which are analytic at μ  =  0. These “nice” paths are characterized by some algebraic equations.

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Authors and Affiliations

  1. The Logistics Institute - Asia Pacific, Block AS6, Level 5, 11 Law Link, Singapore, 119260, Singapore

    Chee-Khian Sim

  2. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Singapore

    Gongyun Zhao

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  1. Chee-Khian Sim
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  2. Gongyun Zhao
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Correspondence to Gongyun Zhao.

Additional information

This research was done during the author’s PhD study at the Department of Mathematics, NUS and as a Research Engineer at the NUS Business School.

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Sim, CK., Zhao, G. Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem. Math. Program. 110, 475–499 (2007). https://doi.org/10.1007/s10107-006-0010-7

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  • DOI: https://doi.org/10.1007/s10107-006-0010-7

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