TY - JOUR
T1 - Superlinear convergence of an infeasible predictor-corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg-Kojima-Monteiro direction
AU - Sim, Chee-Khian
PY - 2011
Y1 - 2011
N2 -
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.
AB -
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.
KW - semidefinite linear complementarity problem
KW - linear semidefinite feasibility problem
KW - interior point method
KW - superlinear convergence
KW - Helmberg-Kojima-Monteiro direction
U2 - 10.1137/090779279
DO - 10.1137/090779279
M3 - Article
SN - 1052-6234
VL - 21
SP - 102
EP - 126
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 1
ER -