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Title: Superlinear convergence of an infeasible predictor-corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg-Kojima-Monteiro direction
Authors: Sim, CK
Issue Date: 2011
Source: SIAM journal on optimization, v. 21, no. 1, p. 102-126
Abstract: An interior point method (IPM) defines a search direction at each interior point of a region. These search directions form a direction field which in turn gives rise to a system of ordinary differential equations (ODEs). The solutions of the system of ODEs can be viewed as underlying paths in the interior of the region. In [C.-K. Sim and G. Zhao, Math. Program. Ser. A, 110 (2007), pp. 475–499], these off-central paths are shown to be well-defined analytic curves, and any of their accumulation points is a solution to a given monotone semidefinite linear complementarity problem (SDLCP). The study of these paths provides a way to understand how iterates generated by an interior point algorithm behave. In this paper, we give a sufficient condition using these off-central paths that guarantees superlinear convergence of a predictor-corrector path-following interior point algorithm for SDLCP using the Helmberg–Kojima–Monteiro (HKM) direction. This sufficient condition is implied by a currently known sufficient condition for superlinear convergence. Using this sufficient condition, we show that for any linear semidefinite feasibility problem, superlinear convergence using the interior point algorithm, with the HKM direction, can be achieved for a suitable starting point. We work under the assumption of strict complementarity.
Keywords: Semidefinite linear complementarity problem
Linear semidefinite feasibility problem
Interior point method
Superlinear convergence
Helmberg–Kojima–Monteiro direction
Publisher: Society for Industrial and Applied Mathematics
Journal: SIAM journal on optimization 
ISSN: 1052-6234
EISSN: 1095-7189
DOI: 10.1137/090779279
Rights: © 2011 Society for Industrial and Applied Mathematics
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