A class of monotone operator equations, which can be decomposed into sum of the gradient of a strongly convex function and a linear and skew-symmetric operator, is considered in this work. Based on discretization of the generalized gradient flow, gradient and skew-symmetric splitting (GSS) methods are proposed and proved to converge in linear rates. To further accelerate the convergence, an accelerated gradient flow is proposed and accelerated gradient and skew-symmetric splitting (AGSS) methods are developed, which extends the acceleration among the existing works on the convex minimization to a more general class of monotone operator equations. In particular, when applied to smooth saddle point systems with bilinear coupling, a linear convergent method with optimal lower iteration complexity is proposed. The robustness and efficiency of GSS and AGSS methods are verified via extensive numerical experiments.
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 effic…
This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach. The proximal point method includes various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers, and thus the proposed acceleration has wide applications. Numerical experiments are presented to demonstrate the accelerating behaviors.
In this paper, based on inertial and Tseng's ideas, we propose two projection-based algorithms to solve a monotone inclusion problem in infinite dimensional Hilbert spaces. Solution theorems of strong convergence are obtained under the certain conditions. Some numerical experiments are presented to illustrate that our algorithms are efficient than the existing results.
In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ is skew. We show its domain is a proper subset of the domain of its adjoint $S^*$, and $-S^*$ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator $T$ on $L^{2}[0,1]$. We compare the domain of $T$ with the domain of its adjoint $T^*$ and show that the skew part of $T$ admits two distinct linear maximal monot…
In this paper, we propose a unifying framework incorporating several momentum-related search directions for solving strongly monotone variational inequalities. The specific combinations of the search directions in the framework are made to guarantee the optimal iteration complexity bound of $\mathcal{O}\left(κ\ln(1/ε)\right)$ to reach an $ε$-solution, where $κ$ is the condition number. This framework provides the flexibility for algorithm designers to train -- among different parameter combinations -- the one that best suits the structure of the problem class at hand. The proposed framework in…
We propose a novel method, namely the accelerated mirror-prox (AMP) method, for computing the weak solutions of a class of deterministic and stochastic monotone variational inequalities (VI). The main idea of this algorithm is to incorporate a multi-step acceleration scheme into the mirror-prox method. For both deterministic and stochastic VIs, the developed AMP method computes the weak solutions with optimal rate of convergence. In particular, if the monotone operator in VI consists of the gradient of a smooth function, the rate of convergence of the AMP method can be accelerated in terms of …
In this technical note, we are concerned with the problem of solving variational inequalities with improved convergence rates. Motivated by Nesterov's accelerated gradient method for convex optimization, we propose a Nesterov's accelerated projected gradient algorithm for variational inequality problems. We prove convergence of the proposed algorithm with at least linear rate under the common assumption of Lipschitz continuity and strongly monotonicity. To the best of our knowledge, this is the first time that convergence of the Nesterov's accelerated protocol is proved for variational inequal…
In this paper, we develop a new type of accelerated algorithms to solve some classes of maximally monotone equations as well as monotone inclusions. Instead of using Nesterov's accelerating approach, our methods rely on a so-called Halpern-type fixed-point iteration in [32], and recently exploited by a number of researchers, including [24, 70]. Firstly, we derive a new variant of the anchored extra-gradient scheme in [70] based on Popov's past extra-gradient method to solve a maximally monotone equation $G(x) = 0$. We show that our method achieves the same $\mathcal{O}(1/k)$ convergence rate (…
Monotone operators are of basic importance in optimization as they generalize simultaneously subdifferential operators of convex functions and positive semidefinite (not necessarily symmetric) matrices. In 1970, Asplund studied the additive decomposition of a maximal monotone operator as the sum of a subdifferential operator and an "irreducible" monotone operator. In 2007, Borwein and Wiersma [SIAM J. Optim. 18 (2007), pp. 946-960] introduced another additive decomposition, where the maximal monotone operator is written as the sum of a subdifferential operator and a "skew" monotone operator. B…
In this paper we provide an algorithm for solving constrained composite primal-dual monotone inclusions, i.e., monotone inclusions in which a priori information on primal-dual solutions is represented via closed convex sets. The proposed algorithm incorporates a projection step onto the a priori information sets and generalizes the method proposed in [Vũ, B.C.: A splitting algorithm for dual monotone inclusions involving cocoercive operators. Adv. Comput. Math. 38, 667-681 (2013)]. Moreover, under the presence of strong monotonicity, we derive an accelerated scheme inspired on [Chambolle, A.; …
We study distributed algorithms for seeking a Nash equilibrium in a class of non-cooperative convex games with strongly monotone mappings. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. To deal with fast distributed learning of Nash equilibria under such settings, we introduce a so called augmented game mapping and provide conditions under which this mapping is strongly monotone. We consider a distributed gradient play algorithm for determining a Nash equilibrium (GRANE). The algorithm involves every player performing…
We address the generalized Nash equilibrium seeking problem for a population of agents playing aggregative games with affine coupling constraints. We focus on semi-decentralized communication architectures, where there is a central coordinator able to gather and broadcast signals of aggregative nature to the agents. By exploiting the framework of monotone operator theory and operator splitting, we first critically review the most relevant available algorithms and then design two novel schemes: (i) a single-layer, fixed-step algorithm with convergence guarantee for general (non cocoercive, non-…
This paper provides a new way of developing the splitting method which is used to solve the problem of finding the resolvent of the sum of maximal monotone operators in Hilbert spaces. By employing accelerated techniques developed by Davis and Yin (in Set-Valued Var. Anal. 25(4):829-858, 2017), this paper presents an implementable, strongly convergent splitting method which is designed to solve the problem. In particular, we show that the distance between the sequence of iterates and the solution converges to zero at a rate O(1/k) to illustrate the efficiency of the proposed method, where k is…
We analyze and characterize maximal monotonicity of linear relations (set-valued operators with linear graphs). An important tool in our study are Fitzpatrick functions. The results obtained partially extend work on linear and at most single-valued operators by Phelps and Simons and by Bauschke, Borwein and Wang. Furthermore, a description of skew linear relations in terms of the Fitzpatrick family is obtained. We also answer one of Simons problems by showing that if a maximal monotone operator has a convex graph, then this graph must actually be affine.
We propose and analyze the convergence of a novel stochastic forward-backward splitting algorithm for solving monotone inclusions given by the sum of a maximal monotone operator and a single-valued maximal monotone cocoercive operator. This latter framework has a number of interesting special cases, including variational inequalities and convex minimization problems, while stochastic approaches are practically relevant to account for perturbations in the data. The algorithm we propose is a stochastic extension of the classical deterministic forward-backward method, and is obtained considering …
We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new {\em Regularized Inertial Proximal Algorithm} obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward-backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possi…
In his recent Proceedings of the AMS paper "Gossez's skew linear map and its pathological maximally monotone multifunctions", Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is nonconvex whenever the scalar is between 0 and 4. The problem of the convexity of that range when the scalar is equal to 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4. We derive this result from an abstract framework that allows us to also obtain a corresponding result for t…
In this paper, we introduce some adaptive methods for solving variational inequalities with relatively strongly monotone operators. Firstly, we focus on the modification of the recently proposed, in smooth case [1], adaptive numerical method for generalized smooth (with Hölder condition) saddle point problem, which has convergence rate estimates similar to accelerated methods. We provide the motivation for such an approach and obtain theoretical results of the proposed method. Our second focus is the adaptation of widespread recently proposed methods for solving variational inequalities with r…
TU Braunschweig（布伦瑞克工业大学）；Graz University of Technology（格拉茨工业大学）；AIT Austrian Institute of Technology GmbH（AIT奥地利技术研究所有限公司）
In this paper, we propose an inertial forward backward splitting algorithm to compute a zero of the sum of two monotone operators, with one of the two operators being co-coercive. The algorithm is inspired by the accelerated gradient method of Nesterov, but can be applied to a much larger class of problems including convex-concave saddle point problems and general monotone inclusions. We prove convergence of the algorithm in a Hilbert space setting and show that several recently proposed first-order methods can be obtained as special cases of the general algorithm. Numerical results show that …
This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strictly convex quadratic functions assuming that exact gradients are used and appropriate upper and lower bounds for the extreme eigenvalues of the Hessian matrix are known. Simple 2-d-examples show that the Euclidean distance of the iterates to the optimal solution is non-monotone. In this context an explicit bound is derived on the number of iterations needed to guarantee a reduction of the Euclidean distance to the optimal solutio…
This paper addresses a distributed convex optimization problem with a class of coupled constraints, which arise in a multi-agent system composed of multiple communities modeled by cliques. First, we propose a fully distributed gradient-based algorithm with a novel operator inspired by the convex projection, called the clique-based projection. Next, we scrutinize the convergence properties for both diminishing and fixed step sizes. For diminishing ones, we show the convergence to an optimal solution under the assumptions of the smoothness of an objective function and the compactness of the cons…
Boston University（波士顿大学）；Northeastern University（东北大学）；CNRS（法国国家科学研究中心）；Université de Paris（巴黎大学）
We provide new adaptive first-order methods for constrained convex optimization. Our main algorithms AdaACSA and AdaAGD+ are accelerated methods, which are universal in the sense that they achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even when they only have access to stochastic gradients. In addition, they do not require any prior knowledge on how the objective function is parametrized, since they automatically adjust their per-coordinate learning rate. These can be seen as truly accelerated Adagrad methods for constrained optimization. We complement them…
The paper proposes a variational-inequality based primal-dual dynamic that has a globally exponentially stable saddle-point solution when applied to solve linear inequality constrained optimization problems. A Riemannian geometric framework is proposed wherein we begin by framing the proposed dynamics in a fiber-bundle setting endowed with a Riemannian metric that captures the geometry of the gradient (of the Lagrangian function). A strongly monotone gradient vector field is obtained by using the natural gradient adaptation on the Riemannian manifold. The Lyapunov stability analysis proves tha…
