  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 set of transition points is finite
and the continuity of the optimal set mapping, on the basis of
Painlev\'e-Kuratowski set convergence, might fail on a nonlinearity interval.
Under a local nonsingularity condition, we then develop 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.
