On the Analyticity of Underlying HKM Paths for Monotone Semidefinite Linear Complementarity Problems.

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 a system of ordinary differential equations (ODEs). The solutions of the system of ODEs are called off-central paths, underlying paths lying in the interior of the feasible region. It is known that not all off-central paths are analytic, whether w.r.t. μ or $\sqrt{\mu}$ , where μ represents the duality gap, at a solution of a given semidefinite linear complementarity problem, SDLCP (Sim and Zhao, Math. Program. 110:475–499, ). In Sim and Zhao (J. Optim. Theory Appl. 137:11–25, ), we give a necessary and sufficient condition for when an off-central path is analytic as a function of $\sqrt{\mu}$ at a solution of a general SDLCP. It is then natural to ask about the analyticity of a SDLCP off-central path at a solution, as a function of μ. We investigate this in the current paper. Again, we work under the assumption that the given SDLCP satisfies strict complementarity condition. [ABSTRACT FROM AUTHOR]