In this paper, we propose a stochastic version of the classical Tseng's forward-backward-forward method with inertial term for solving monotone inclusions given by the sum of a maximal monotone operator and a single-valued monotone operator in real Hilbert spaces. We obtain the almost sure convergence for the general case and the rate $\mathcal{O}(1/n)$ in expectation for the strong monotone case.
Furthermore, we derive $\mathcal{O}(1/n)$ rate convergence of the primal-dual gap for saddle point problems.
In this paper, based on a double inertial extrapolation steps strategy and relaxation techniques, we introduce a new Tseng splitting method with double inertial extrapolation steps and self-adaptive step sizes for solving monotone inclusion problems in real Hilbert spaces. Finally, several numerical experiments are provided to illustrate the performance and theoretical outcomes of our algorithm.
In this paper, we propose a variable metric version of Tseng's algorithm (the forward-backward-forward algorithm: FBF) combined with extrapolation from the past that includes error terms for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. This can be seen as the optimistic gradient descent ascent (OGDA) algorithm endowed with variable metrics and error terms. Primal-dual algorithms are also proposed for monotone inclusion problems involving compositions with linear operators. The primal-dual problem occurring in image deblurrin…
Iteration complexity of an inexact Douglas-Rachford method and of a Douglas-Rachford-Tseng's F-B four-operator splitting method for solving monotone inclusions
Forward-Backward and Tseng's Type Penalty Schemes for Monotone Inclusion Problems
研究实希尔伯特空间中$0\in Ax+Dx+N_C(x)$形式的单调包含问题，其中$A$是极大单调算子，$D$是余强制算子，$C$是余强制算子的非空零点集.我们提出了一个求解该问题的前向-后向惩罚算法，它推广了H。Attouch，M.- O. Czarnecki和J. Peypouquet在[3]中。该条件保证弱遍历收敛的迭代序列所产生的建议计划制定的Fitzpatrick功能相关联的最大单调算子，描述了集$C$。在第二部分中，我们介绍了一个向前向后向前算法的单调包含问题具有相同的结构，但这一次通过取代的协尾假设与Lipschitz连续性条件。后一种惩罚型算法为处理结构更复杂的单调包含问题打开了大门，例如，涉及极大单调算子与线性连续算子的复合。
Tseng's forward-backward-forward algorithm is a valuable alternative for Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed by monotone and Lipschitz continuous operators, as it requires in every step only one projection operation. However, it is well-known that Korpelevich's method converges and can therefore be used also for solving variational inequalities governed by pseudo-monotone and Lipschitz continuous operators. In this paper, we first associate to a pseudo-monotone variational inequality a forward-backward-forward dynamical…
Forward-backward-forward methods with variance reduction for stochastic variational inequalities
University of Vienna（维也纳大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；Grenoble INP（格勒诺布尔国立理工学院）；LIG（格勒诺布尔信息学实验室）；Maastricht University（马斯特里赫特大学）
Motivated by the training of Generative Adversarial Networks (GANs), we study methods for solving minimax problems with additional nonsmooth regularizers. We do so by employing \emph{monotone operator} theory, in particular the \emph{Forward-Backward-Forward (FBF)} method, which avoids the known issue of limit cycling by correcting each update by a second gradient evaluation. Furthermore, we propose a seemingly new scheme which recycles old gradients to mitigate the additional computational cost. In doing so we rediscover a known method, related to \emph{Optimistic Gradient Descent Ascent (OGD…
We extend the Malitsky-Tam forward-reflected-backward (FRB) splitting method for inclusion problems of monotone operators to nonconvex minimization problems. By assuming the generalized concave Kurdyka-Łojasiewicz (KL) property of a quadratic regularization of the objective, we show that the FRB method converges globally to a stationary point of the objective and enjoys finite length property. The sharpness of our approach is guaranteed by virtue of the exact modulus associated with the generalized concave KL property. Numerical experiments suggest that FRB is competitive compared to the Dougl…
In this article we provide a splitting method for solving monotone inclusions in a real Hilbert space involving four operators: a maximally monotone, a monotone-Lipschitzian, a cocoercive, and a monotone-continuous operator. The proposed method takes advantage of the intrinsic properties of each operator, generalizing the forward-back-half forward splitting and the Tseng's algorithm with line-search. At each iteration, our algorithm defines the step-size by using a line search in which the monotone-Lipschitzian and the cocoercive operators need only one activation. We also derive a method for …
Monotone inclusions involving the sum of three maximally monotone operators or more have received much attention in recent years. In this paper, we propose three splitting algorithms for finding a zero of the sum of four monotone operators, which are two maximally monotone operators, one monotone Lipschitz operator, and one cocoercive operator. These three splitting algorithms are based on the forward-reflected-Douglas-Rachford splitting algorithm, backward-forward-reflected-backward splitting algorithm, and backward-reflected-forward-backward splitting algorithm, respectively. As applications…
Monotone inclusions have wide applications in solving various convex optimization problems arising in signal and image processing, machine learning, and medical image reconstruction. In this paper, we propose a new splitting algorithm for finding a zero of the sum of a maximally monotone operator, a monotone Lipschitzian operator, and a cocoercive operator, which is called outer reflected forward-backward splitting algorithm. Under mild conditions on the iterative parameters, we prove the convergence of the proposed algorithm. As applications, we employ the proposed algorithm to solve composit…
Splitting Algorithms Using Pairs Monotonicity to Solve the Sum of Two Non-monotone Operators Problem
Ton Duc Thang University（胡志明市镏德胜大学）；Sidi Mohammed Ben Abdellah University, National School of Applied Sciences（西迪穆罕默德·本阿卜达拉大学国立应用科学学院）；University of Limoges（利摩日大学）
In this paper, we propose and study several strongly convergent versions of the forward-reflected-backward splitting method of Malitsky and Tam for finding a zero of the sum of two monotone operators in a real Hilbert space. Our proposed methods only require one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. We also develop inertial versions of our methods and strong convergence results are obtained for these metho…
In this paper, we develop rapidly convergent forward-backward algorithms for computing zeroes of the sum of finitely many maximally monotone operators. A modification of the classical forward-backward method for two general operators is first considered, by incorporating an inertial term (closed to the acceleration techniques introduced by Nesterov), a constant relaxation factor and a correction term. In a Hilbert space setting, we prove the weak convergence to equilibria of the iterates $(x_n)$, with worst-case rates of $ o(n^{-2})$ in terms of both the discrete velocity and the fixed point r…
It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone op…
This paper investigates first-order variable metric backward forward dynamical systems associated with monotone inclusion and convex minimization problems in real Hilbert space. The operators are chosen so that the backward-forward dynamical system is closely related to the forward-backward dynamical system and has the same computational complexity. We show existence, uniqueness, and weak asymptotic convergence of the generated trajectories and strong convergence if one of the operators is uniformly monotone. We also establish that an equilibrium point of the trajectory is globally exponential…
We study stochastic Nash equilibrium problems with expected valued cost functions whose pseudogradient satisfies restricted monotonicity properties which hold only with respect to the solution. We propose a forward-backward algorithm and prove its convergence under restricted strong monotonicity, restricted strict monotonicity and restricted cocoercivity of the pseudogradient mapping. To approximate the expected value, we use either a finite number of samples and a vanishing step size or an increasing number of samples with a constant step. Numerical simulations show that our proposed algorith…
We propose a novel dynamically weighted inertial forward-backward algorithm (DWIFOB) for solving structured monotone inclusion problems. The scheme exploits the globally convergent forward-backward algorithm with deviations in [26] as the basis and combines it with the extrapolation technique used in Anderson acceleration to improve local convergence. We also present a globally convergent primal-dual variant of DWIFOB and numerically compare its performance to the primal-dual method of Chambolle-Pock and a Tikhonov regularized version of Anderson acceleration applied to the same problem. In al…
We propose and study a weakly convergent variant of the forward--backward algorithm for solving structured monotone inclusion problems. Our algorithm features a per-iteration deviation vector which provides additional degrees of freedom. The only requirement on the deviation vector to guarantee convergence is that its norm is bounded by a quantity that can be computed online. This approach provides great flexibility and opens up for the design of new and improved forward--backward-based algorithms, while retaining global convergence guarantees. These guarantees include linear convergence of ou…
Strong Convergence of Forward-Backward-Forward Methods for Pseudo-monotone Variational Inequalities with Applications to Dynamic User Equilibrium in Traffic Networks
We propose and analyze the convergence of a novel stochastic algorithm for solving monotone inclusions that are the sum of a maximal monotone operator and a monotone, Lipschitzian operator. The propose algorithm requires only unbiased estimations of the Lipschitzian operator. We obtain the rate $\mathcal{O}(log(n)/n)$ in expectation for the strongly monotone case, as well as almost sure convergence for the general case. Furthermore, in the context of application to convex-concave saddle point problems, we derive the rate of the primal-dual gap. In particular, we also obtain $\mathcal{O}(1/n)$ …
Coupled Adaptable Backward-Forward-Backward Resolvent Splitting Algorithm (CABRA): A Matrix-Parametrized Resolvent Splitting Method for the Sum of Maximal Monotone and Cocoercive Operators Composed with Linear Coupling Operators
United States Naval Academy（美国海军学院）；Naval Postgraduate School（海军研究生院）
