  The breakthrough ideas in the modern proximal splitting methodologies allow
us to express the set of all minimizers of a superposition of multiple
nonsmooth convex functions as the fixed point set of computable nonexpansive
operators. In this paper, we present practical algorithmic strategies for the
hierarchical convex optimization problems which require further strategic
selection of a most desirable vector from the solution set of the standard
convex optimization. The proposed algorithms are established by applying the
hybrid steepest descent method to special nonexpansive operators designed
through the art of proximal splitting. We also present applications of the
proposed strategies to certain unexplored hierarchical enhancements of the
support vector machine and the Lasso estimator.
