  This study addresses some algorithms for solving structured unconstrained
convex optimiza- tion problems using first-order information where the
underlying function includes high-dimensional data. The primary aim is to
develop an implementable algorithmic framework for solving problems with multi-
term composite objective functions involving linear mappings using the optimal
subgradient algorithm, OSGA, proposed by Neumaier in [49]. To this end, we
propose some prox-functions for which the cor- responding subproblem of OSGA is
solved in a closed form. Considering various inverse problems arising in signal
and image processing, machine learning, statistics, we report extensive
numerical and compar- isons with several state-of-the-art solvers proposing
favourably performance of our algorithm. We also compare with the most widely
used optimal first-order methods for some smooth and nonsmooth con- vex
problems. Surprisingly, when some Nesterov-type optimal methods originally
proposed for smooth problems are adapted for solving nonsmooth problems by
simply passing a subgradient instead of the gradient, the results of these
subgradient-based algorithms are competitive and totally interesting for
solving nonsmooth problems. Finally, the OSGA software package is available.
