Stabilizing ill-conditioned linear complementarity problems

Least-squares adjustment with inequality constraints is equivalent to solving a linear complementarity problem (LCP) with a positive definite matrix; the latter has, however, received little attention in geodesy. Two kinds of LCP solution methods (direct and approximation) have been analysed from the point of view of solution stability. It has been found that they can result in an unreliable solution to an unstable LCP with a positive definite matrix, the extent of which depends on the corresponding submatrix. Several proposals for improving the solutions to the unstable LCP are suggested. Two examples are given.