Asymptotic Behavior of HKM Paths in Interior Point Methods for Monotone Semidefinite Linear Complementarity Problems: General Theory

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 [9], these offcentral 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 of a simple example are also studied in [9] whose asymptotic behavior near the solution of the example is analyzed. In this paper, which is an extension of [9], we study the asymptotic behavior of off-central paths for general SDLCPs (using the dual HKM direction), instead of for a given example. We give a necessary and sufficient condition for when an off-central path is analytic as a function of √ μ 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 √ μ, is bounded. We work under the assumption that the given SDLCP satisfies strict complementarity condition.