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We investigate the techniques and ideas used in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration we allow the use of variable metrics, while the investigations are carried out in the setting of infinite dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties., Comment: Updates in Section 2 concerning the derivation of the convergence rates + a unifying convergence theorem for the sequence of iterates

Let A be a selfadjoint operator and P be an orthogonal projection both operating on a Hilbert space script H sign. We say that A is P-complementable if A-μP≥0 holds for some μ∈R. In this case we define I P(A)=max{μ∈R:A-μP≥0}. As a tool for computing I P(A) we introduce a natural generalization of the Schur complement or shorted operator of A to script S sign=R(P), denoted by Σ(A,P). We give expressions and a characterization for I P(A) that generalize some known results for particular choices of P. We also study some aspects of the shorted operator Σ(A,P) for P-complementable A, under the hypothesis of compatibility of the pair (A,script S sign). We give some applications in the finite dimensional context., Facultad de Ciencias Exactas