  Composite minimization involves a collection of smooth functions which are
aggregated in a nonsmooth manner. In the convex setting, we design an algorithm
by linearizing each smooth component in accordance with its main curvature. The
resulting method, called the Multiprox method, consists in solving successively
simple problems (e.g. constrained quadratic problems) which can also feature
some proximal operators. To study the complexity and the convergence of this
method we are led to study quantitative qualification conditions to understand
the impact of multipliers on the complexity bounds. We obtain explicit
complexity results of the form $O(\frac{1}{k})$ involving new types of constant
terms. A distinctive feature of our approach is to be able to cope with oracles
involving moving constraints. Our method is flexible enough to include the
moving balls method, the proximal Gauss-Newton's method, or the
forward-backward splitting, for which we recover known complexity results or
establish new ones. We show through several numerical experiments how the use
of multiple proximal terms can be decisive for problems with complex
geometries.
