Optimal subgradient methods: computational properties for large-scale linear inverse problems

This paper studies the computational properties of the optimal subgradient algorithm (OSGA) for applications of linear inverse problems involving high-dimensional data. First, such convex problems are formulated as a class of convex problems with multi-term composite objective functions involving linear mappings. Next, an efficient procedure for computing the first-order oracle for such problems is provided and OSGA is equipped with some prox-functions such that the OSGA subproblem is solved in a closed form. Further, a comprehensive comparison among the most popular first-order methods is given. Then, several Nesterov-type optimal methods (originally proposed for smooth problems) are adapted to solve nonsmooth problems by simply passing a subgradient instead of the gradient, where the results of these subgradient methods are competitive and totally interesting for solving nonsmooth problems. Finally, numerical results with several inverse problems (deblurring with isotropic total variation, elastic net, and ℓ1-minimization) show the efficiency of OSGA and the adapted Nesterov-type optimal methods for large-scale problems. For the deblurring problem, the efficiency measures of the improvement on the signl-to-noise ratio and the peak signal-to-noise ratio are used. The software package implementing OSGA is publicly available.© The Author(s) 201