A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality

Abstract

The principle underlying this paper is the basic observation that the problem of simultaneously solving a large class of composite monotone inclusions and their duals can be reduced to that of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator. An algorithmic framework is developed for solving this generic problem in a Hilbert space setting. New primal-dual splitting algorithms are derived from this framework for inclusions involving composite monotone operators, and convergence results are established. These algorithms draw their simplicity and efficacy from the fact that they operate in a fully decomposed fashion in the sense that the monotone operators and the linear transformations involved are activated separately at each iteration. Comparisons with existing methods are made and applications to composite variational problems are demonstrated.2000 Mathematics Subject Classification: Primary 47H05; Secondary 65K05, 90C25.

A Primal–Dual Splitting Method for Convex Optimization Involving Lipschitzian, Proximable and Linear Composite Terms

“…Thus, Theorems 3.2 and 3.3 extend the domain of convergence of these primal-dual algorithms. When F = 0, the primal-dual method in [19] and the method in [38] can be used to solve (1), as well. They yield algorithms different from Algorithms 3.1 and 3.2.…”

“…In return, this quest of practitioners for efficient minimization methods has caused a renewed interest among mathematicians around splitting methods in monotone and nonexpansive operator theory, as can be judged from the numerous recent contributions, e.g. [13][14][15][16][17][18][19][20][21][22][23][24]. The most classical operator splitting methods to minimize the sum of two convex functions are the forward-backward method, proposed in [2] and further developed in [3,4,7,[25][26][27][28], and the Douglas-Rachford method [3,6,7,22].…”

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

“…On the other hand, since, due to Proposition 3.5, Algorithm 2 is a special instance of Algorithm 1, it converges for all σ ∈ [0, 1[ (see, e.g., [3,Theorem 3.1]). Note that the difference between sPIM(ε) and Algorithm 2 is the inequality in (29) and (31).…”

“…(51) Due to its importance in solving large-scale problems, numerical schemes for solving (49) use information of each T i individually instead of using the entire sum [1,17,27,29,30,31]. In this section, we apply the results of Section 3 to present and study the iteration-complexity of an inexact-version of the Spingarn's operator splitting method [1] for solving (49) and, as a by-product, we obtain the iteration-complexity of the latter method.…”

Sparse point-source removal for full-sky CMB experiments: application to WMAP 9-year data

“…(2) can be solved using primal-dual approaches (e.g. Chambolle & Pock 2011;Briceño Arias & Combettes 2011;Becker et al 2011;Combettes & Pesquet 2012). The proposed algorithm was derived from Chambolle & Pock (2011), it requires only one application of the costly spherical harmonic and wavelet transforms and one application of their adjoint per iteration and does not require subiterations.…”