A product space reformulation with reduced dimension for splitting algorithms

Abstract

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.

A primal-dual splitting algorithm for composite monotone inclusions with minimal lifting

“…Remark 2 (On the parameter γ in the definition of norm • γ ) In Lemma 2, we proved that the operator T is λ-averaged nonexpansive with respect to the norm • γ induced by the scalar product defined in (8). Although the use of this norm did not require detours from the usual procedure to prove convergence of the fixed point algorithm in Theorem 3, it may numerically affect the performance of the algorithm.…”

“…Until very recently, the only way to tackle the problem when n > 2 was using Pierra's product space reformulation [18], which implies an n-fold lifting. Nowadays, various algorithms have been proposed allowing to solve the problem by only resorting to an (n − 1)-lifting (see, e.g., [8,13,14]). This reduction from n to n − 1 has been proven to be minimal [16] when the algorithms are required to be frugal resolvent splittings [19], which means that each of the resolvents J A 1 , .…”

“…, A n is used exactly once. Example 1 In [8], a product space reformulation with reduced dimension is proposed, which applied to Problem 2 yields the following lifted splitting. Given any γ > 0 and λ ∈ ]0, 2], the algorithm in [8, Theorem 5.1] can be defined by the operator R :…”

“…Recalling Remark 2, it is of interest to consider a mechanism which allows tuning the parameter γ appearing in the definition of the norm given by the inner product in (8) to an appropriate value. To this aim, we perform in (31) a change of variable of the form s = μx, with μ > 0, and instead handle the problem…”

“…In general, a smaller dimension of the space translates into less consumption of computational resources. For this reason, the development of algorithms with reduced dimension for solving monotone inclusion problems has recently become an active topic of research [8,14,16,19].…”