Hierarchical Convex Optimization With Primal-Dual Splitting

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.