Recent Developments on Primal–Dual Splitting Methods with Applications to Convex Minimization

Abstract

This chapter presents a survey on primal-dual splitting methods for solving monotone inclusion problems involving maximally monotone operators, linear compositions of parallel sums of maximally monotone operators, and singlevalued Lipschitzian or cocoercive monotone operators. The primal-dual algorithms have the remarkable property that the operators involved are evaluated separately in each iteration, either by forward steps in the case of the single-valued ones or by backward steps for the set-valued ones, by using the corresponding resolvents. In the hypothesis that strong monotonicity assumptions for some of the involved operators are fulfilled, accelerated algorithmic schemes are presented and analyzed from the point of view of their convergence. Finally, we discuss the employment of the primal-dual methods in the context of solving convex optimization problems arising in the fields of image denoising and deblurring, support vector machine learning, location theory, portfolio optimization and clustering.

Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators

“…In this section, we assume that there exists x such that x ∈ zer (D + E + F ). This assumption yields that the set of primal-dual solutions is nonempty (see [3,11] and the references therein for more discussion). Reformulate (56) in the form of (3) by defining…”

“…This approach yields the primal and dual solutions simultaneously (hence the name primal-dual splittings) and eliminates the need to calculate the proximal mapping of a linearly composed function. The resulting algorithms only require matrix vector products, gradient and proximal updates (see [3,8,11,14,36] for more discussion). We follow the same approach and notice that it is quite natural to embed the optimality condition of the saddle point problem associated to (2) in the form of the monotone inclusion (1).…”

“…The sequence (xn) n∈IN converges to x R-linearly if there is a sequence of nonnegative scalars (vn) n∈IN such that xn − x ≤ vn and (vn) n∈IN converges Q-linearly 3 to zero 3. The sequence (xn) n∈IN converges to x Q-linearly with Q-factor given by σ ∈]0, 1[, if for n sufficiently large x n+1 − x ≤ σ xn − x holds.…”

Distributed Proximal Splitting Algorithms with Rates and Acceleration

“…Many estimation problems in a wide range of scientific fields can be formulated as large-scale convex optimization problems (Palomar and Eldar, 2009;Sra et al, 2011;Bach et al, 2012;Bubeck, 2015;Polson et al, 2015;Chambolle and Pock, 2016a;Glowinski et al, 2016;Stathopoulos et al, 2016;Condat, 2017a;Condat et al, 2019b). Proximal splitting algorithms (Combettes and Pesquet, 2010;Boţ et al, 2014;Parikh and Boyd, 2014;Komodakis and Pesquet, 2015;Beck, 2017;Condat et al, 2019a) are particularly well suited to solve them; they consist of simple, easy to compute, steps that can deal with the terms in the objective function separately.…”

Phase retrieval problems in x-ray physics

“…FISTA [11], ADMM [18], proximal splitting algorithms [25]. We also commend [15,61] for an overview. However, these methods usually assume X and Y n to be Hilbert spaces.…”