Existence and uniqueness conditions for the dual linear complementarity problem

Characterize conditions guaranteeing the existence and uniqueness of solutions to the linear complementarity problem used to compute the optimal dual variable for constrained finite-horizon linear quadratic Gaussian games, particularly when the matrix -\tilde C is not readily verifiable as a P-matrix.

Background

The constrained inverse LQG-game procedure computes optimal shared dual variables by solving a linear complementarity problem involving the matrix $\tilde C$. A sufficient condition for existence and uniqueness is that $-\tilde C$ be a P-matrix, but the paper notes that verifying this property is difficult because $-\tilde C$ is generally neither symmetric nor sign-definite. Numerical experiments indicate that the problem often has a solution even when $-\tilde C$ is not positive definite, leaving general existence and uniqueness conditions unresolved.