On computing the nonlinearity interval in parametric semidefinite optimization.

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the continuity of the optimal set mapping, on the basis of Painleve-Kuratowski set convergence, might fail on a nonlinearity interval. Furthermore, under a mild assumption, we prove that the set of transition points and the set of points at which the optimal set mapping is discontinuous are finite. We then present a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.