General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems

Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com-plementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (1998), this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of Monteiro and Pang (1998) to this general map. Each fundamental map leads to a family of “weighted” continuous trajectories which include the central trajectory as a special case. Hence, for complementarity problems over the cone of symmetric positive semidefinite matrices, the notion of weighted central path depends on the fundamental map used to represent the central path.