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

An interior point method (IPM) 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 √ μ 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.