An Inexact Spingarn’s Partial Inverse Method with Applications to Operator Splitting and Composite Optimization

Abstract

We propose and study the iteration-complexity of an inexact version of the Spingarn's partial inverse method. Its complexity analysis is performed by viewing it in the framework of the hybrid proximal extragradient (HPE) method, for which pointwise and ergodic iteration-complexity has been established recently by Monteiro and Svaiter. As applications, we propose and analyze the iteration-complexity of an inexact operator splitting algorithm -which generalizes the original Spingarn's splitting method -and of a parallel forward-backward algorithm for multi-term composite convex optimization.

Tikhonov-like methods with inexact minimization for solving linear ill-posed problems

“…for all λ ∈ (0, 1) and for all x, y ∈ dom( f ). The function ϕ is called the convexity module of f. If (10) holds with ϕ(•) = (β/2)(•) 2 for some β > 0, then f is called strongly convex with constant β. Of course f uniformly convex implies f strictly convex, which in turn implies f convex.…”

“…If f is strongly convex with constant β, then the convexity module is given by ϕ(•) = (β/2)(•) 2 . In this case, inequality (14) reads D ν (x, y) (β/2) x − y 2 and lemma 1 implies that ξ − ν, x − y β x − y 2 , for all ξ ∈ ∂ f(x) and ν ∈ ∂ f(y).…”

“…where A : X → Y is a bounded linear operator, whose inverse A −1 : R(A) → X either does not exist or is not continuous. In practice, one does not know the data b exactly, and only approximate measured data b δ ∈ Y are available, with b − b δ δ, ( 2 ) where δ > 0 is the (known) noise level. Tikhonov regularization is probably the most popular regularization technique for producing stable solutions of (1) from (2).…”

“…In practice, one does not know the data b exactly, and only approximate measured data b δ ∈ Y are available, with b − b δ δ, ( 2 ) where δ > 0 is the (known) noise level. Tikhonov regularization is probably the most popular regularization technique for producing stable solutions of (1) from (2). The most classical formulation of Tikhonov method consists in finding the (unique) minimizer x α ∈ X of the convex functional…”

On the convergence rate of the scaled proximal decomposition on the graph of a maximal monotone operator (SPDG) algorithm

“…Among the above mentioned measures of approximate solution, one of them allows for the study of the iteration-complexity of the SPDG algorithm in the lines of recent results on the iterationcomplexity of the inexact Spingarn's partial inverse method [2] (see (33)). In this regard, one can compute SPDG algorithm's iteration-complexity with respect to the following notion of approximate solution of (8) (see [2]): for a given tolerance ρ > 0, find x, u ∈ H such that…”