Underlying Paths in Interior Point Method for Monotone Semidefinite Linear Complementarity Problem

An interior point method (IPM) defines a search direction at an interior point of the feasible region. These search directions form a direction field which in turn defines, as the unique solution of a system of ordinary differential equations (ODEs), a curve passing through any interior point. We call such curves off-central paths (the central path is trivial for analysis; our concern is off-central paths, so the name). We study off-central paths for monotone semidefinite linear complementarity problem (SDLCP). We show that each off-central path is a welldefined 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 whose first derivatives 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. Email: [email protected] 355