  We consider rather a general class of multi-level optimization problems,
where a convex objective function is to be minimized subject to constraints of
optimality of nested convex optimization problems. As a special case, we
consider a trilevel optimization problem, where the objective of the two lower
layers consists of a sum of a smooth and a non-smooth term.~Based on
fixed-point theory and related arguments, we present a natural first-order
algorithm and analyze its convergence and rates of convergence in several
regimes of parameters.
