This paper addresses the selection of a desirable solution
among all the solutions of a convex optimization problem (referred
to as the first-stage problem) mainly for inverse problems in
signal processing. This is realized in the framework of hierarchical
convex optimization, i.e., minimizing another convex function over
the solution set of the first-stage problem. Hierarchical convex optimization
is an ideal strategy when the first-stage problem has infinitely
many solutions because of the non-strict convexity of its objective
function, which could arise in various scenarios, e.g., convex
feasibility problems. To this end, first, the fixed point set characterization
behind a primal-dual splitting type method is incorporated
into the framework of hierarchical convex optimization, which enables
the framework to cover a broad class of first-stage problem
formulations. Then, a pair of efficient algorithmic solutions to the
hierarchical convex optimization problem, as certain realizations
of the hybrid steepest descent method, are provided with guaranteed
convergence. We also present a specialized form of the proposed
framework to focus on a typical scenario of inverse problems,
and show its application to signal interpolation.
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