We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient...

We introduce an axiomatic formalism for the concept of the center of a set in a Euclidean space. Then we explain how to exploit possible symmetries and possible cyclicities in the set in order to localize its center. Special attention is paid to the determination of centers in cones of matrices....

In this paper, we develop new subgradient methods for solving nonsmooth convex optimization problems. These methods are the first ones, for which the whole sequence of test points is endowed with the worst-case performance guarantees. The new methods are derived from a relaxed estimating...

We consider a class of nonsmooth convex optimization problems where the objective function is the composition of a strongly convex differentiable function with a linear mapping, regularized by the group reproducing kernel norm. This class of problems arise naturally from applications in group...