  In this work, we consider methods for solving large-scale optimization
problems with a possibly nonsmooth objective function. The key idea is to first
specify a class of optimization algorithms using a generic iterative scheme
involving only linear operations and applications of proximal operators. This
scheme contains many modern primal-dual first-order solvers like the
Douglas-Rachford and hybrid gradient methods as special cases. Moreover, we
show convergence to an optimal point for a new method which also belongs to
this class. Next, we interpret the generic scheme as a neural network and use
unsupervised training to learn the best set of parameters for a specific class
of objective functions while imposing a fixed number of iterations. In contrast
to other approaches of "learning to optimize", we present an approach which
learns parameters only in the set of convergent schemes. As use cases, we
consider optimization problems arising in tomographic reconstruction and image
deconvolution, and in particular a family of total variation regularization
problems.
