  In this paper, we propose a new Fully Composite Formulation of convex
optimization problems. It includes, as a particular case, the problems with
functional constraints, max-type minimization problems, and problems of
Composite Minimization, where the objective can have simple nondifferentiable
components. We treat all these formulations in a unified way, highlighting the
existence of very natural optimization schemes of different order. We prove the
global convergence rates for our methods under the most general conditions.
Assuming that the upper-level component of our objective function is
subhomogeneous, we develop efficient modification of the basic Fully Composite
first-order and second-order Methods, and propose their accelerated variants.
