A global optimization approach for solving non-monotone equilibrium problems (EPs) is proposed. The class of (regularized) gap functions is used to reformulate any EP as a constrained global optimization program and some bounds on the Lipschitz constant of such functions are provided. The proposed global optimization approach is a combination of an improved version of the \texttt{DIRECT} algorithm, which exploits local bounds of the Lipschitz constant of the objective function, with local minimizations. Unlike most existing solution methods for EPs, no monotonicity-type condition is assumed in this paper. Preliminary numerical results on several classes of EPs show the effectiveness of the approach.
Basing on some recently proposed methods for solving variational inequalities with non-smooth operators, we propose an analogue of the Mirror Prox method for the corresponding class of problems under the assumption of relative smoothness and relative strong monotonicity of the operator.
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.
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…
In this paper we provide a splitting method for finding a zero of the sum of a maximally monotone operator, a lipschitzian monotone operator, and a normal cone to a closed vectorial subspace of a real Hilbert space. The problem is characterized by a simpler monotone inclusion involving only two operators: the partial inverse of the maximally monotone operator with respect to the vectorial subspace and a suitable lipschitzian monotone operator. By applying the Tseng's method in this context we obtain a splitting algorithm that exploits the whole structure of the original problem and generalizes…
In this work we propose a new splitting technique, namely Asymmetric Forward-Backward-Adjoint splitting, for solving monotone inclusions involving three terms, a maximally monotone, a cocoercive and a bounded linear operator. Classical operator splitting methods, like Douglas-Rachford and Forward-Backward splitting are special cases of our new algorithm. Asymmetric Forward-Backward-Adjoint splitting unifies, extends and sheds light on the connections between many seemingly unrelated primal-dual algorithms for solving structured convex optimization problems proposed in recent years. More import…
This paper studies the iteration-complexity of new regularized hybrid proximal extragradient (HPE)-type methods for solving monotone inclusion problems (MIPs). The new (regularized HPE-type) methods essentially consist of instances of the standard HPE method applied to regularizations of the original MIP. It is shown that its pointwise iteration-complexity considerably improves the one of the HPE method while approaches (up to a logarithmic factor) the ergodic iteration-complexity of the latter method.
In this paper we provide a splitting algorithm for solving coupled monotone inclusions in a real Hilbert space involving the sum of a normal cone to a vector subspace, a maximally monotone, a monotone-Lipschitzian, and a cocoercive operator. The proposed method takes advantage of the intrinsic properties of each operator and generalizes the method of partial inverses and the forward-backward-half forward splitting, among other methods. At each iteration, our algorithm needs two computations of the Lipschitzian operator while the cocoercive operator is activated only once. By using product spac…
We introduce the first operator splitting method for composite monotone inclusions outside of Hilbert spaces. The proposed primal-dual method constructs iteratively the best Bregman approximation to an arbitrary point from the Kuhn-Tucker set of a composite monotone inclusion. Strong convergence is established in reflexive Banach spaces without requiring additional restrictions on the monotone operators or knowledge of the norms of the linear operators involved in the model. The monotone operators are activated via Bregman distance-based resolvent operators. The method is novel even in Euclide…
We propose a new class of primal-dual Fejer monotone algorithms for solving systems of com- posite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the model are used to con- struct a half-space containing the Kuhn-Tucker set associated with the system. The primal-dual update is then obtained via a relaxed projection of the current iterate onto this half-space. An important feature that distinguish…
We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monot…
We propose two very simple methods, the first one with constant step sizes and the second one with self-adaptive step sizes, for finding a zero of the sum of two monotone operators in real reflexive Banach spaces. Our methods require only one evaluation of the single-valued operator at each iteration. Weak convergence results are obtained when the set-valued operator is maximal monotone and the single-valued operator is Lipschitz continuous, and strong convergence results are obtained when either one of these two operators is required, in addition, to be strongly monotone. We also obtain the r…
We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic structure of the problem by activating explicitly the cocoercive operator in the first step, and taking advantage of a vector space decomposition in the second step. The second step of the first method is a Douglas-Rachford iteration involving the maximally monotone operator and the normal cone. In the second method it is a proximal step involving the partial …
In this paper, we introduce an inertial proximal method for solving a bilevel problem involving two monotone equilibrium bifunctions in Hilbert spaces. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the iterative method are established. Two particular cases illustrating the proposed method are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under saddle point constraint. Furthermore, a numerical example is given to demonstrate the implementability…
In this work, we study the distributed Nash equilibrium seeking problem for monotone generalized noncooperative games with set constraints and shared affine inequality constraints. A distributed regularized penalty method is proposed. The idea is to use a differentiable penalty function with a time-varying penalty parameter to deal with the inequality constraints. A time-varying regularization term is used to deal with the ill-poseness caused by the monotonicity assumption and the time-varying penalty term. The asymptotic convergence to the least-norm variational equilibrium of the game is pro…
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 propose a splitting method for solving an equilibrium problem involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of two appropriate maximally monotone operators. Our algorithm is a consequence of the Douglas--Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.
We propose a new primal-dual splitting method for solving composite inclusions involving Lipschitzian, and parallel-sum-type monotone operators. Our approach extends the framework in \cite{Siopt4} to a more general class of monotone inclusions in a nontrivial fashion. The main idea is to represent the solution set of both the primal and dual problems using their associated Kuhn-Tucker set, and then develop a projected method to successively approximate a feasible point of the Kuhn-Tucker set. We propose a splitting algorithm based on the resolvent of each maximally monotone operator to constru…
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-…
The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some elaborated splitting techniques, all of the operators occurring in the problem formulation are processed individually via forward or backward steps. The treatment of parallel sums of linearly composed maximally monotone operators is motivated by applications in imaging which involve first- and second-order total variation functionals, to which a special attentio…
Tseng's algorithm finds a zero of the sum of a maximally monotone operator and a monotone continuous operator by evaluating the latter twice per iteration. In this paper, we modify Tseng's algorithm for finding a zero of the sum of three operators, where we add a cocoercive operator to the inclusion. Since the sum of a cocoercive and a monotone-Lipschitz operator is monotone and Lipschitz, we could use Tseng's method for solving this problem, but implementing both operators twice per iteration and without taking into advantage the cocoercivity property of one operator. Instead, in our approach…
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…
The standard algorithms for solving large-scale convex-concave saddle point problems, or, more generally, variational inequalities with monotone operators, are proximal type algorithms which at every iteration need to compute a prox-mapping, that is, to minimize over problem's domain $X$ the sum of a linear form and the specific convex distance-generating function underlying the algorithms in question. Relative computational simplicity of prox-mappings, which is the standard requirement when implementing proximal algorithms, clearly implies the possibility to equip $X$ with a relatively comput…
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…
Deep equilibrium models are based on implicitly defined functional relations and have shown competitive performance compared with the traditional deep networks. Monotone operator equilibrium networks (monDEQ) retain interesting performance with additional theoretical guaranties. Existing certification tools for classical deep networks cannot directly be applied to monDEQs for which much fewer tools exist. We introduce a semialgebraic representation for ReLU based monDEQs which allows to approximate the corresponding input output relation by semidefinite programming (SDP). We present several ap…
It is well known that the projection method is not convergent for monotone equilibrium problems. Recently Sosa \textit{et al.} in \cite{SS2011} proposed a projection algorithm ensuring convergence for paramonotone equilibrium problems. In this paper we modify this algorithm to obtain a splitting convergent one for the case when the bifunction is the sum of the two ones. At each iteration, two strongly convex subprograms are required to solve separately, one for each component bifunction. We show that the algorithm is convergent for paramonotone bifunction without any Lipschitz type condition a…
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…
In this paper we propose new algorithms for solving a class of structured monotone variational inequality (VI) problems over compact feasible sets. By identifying the gradient components existing in the operator of VI, we show that it is possible to skip computations of the gradients from time to time, while still maintaining the optimal iteration complexity for solving these VI problems. Specifically, for deterministic VI problems involving the sum of the gradient of a smooth convex function $\nabla G$ and a monotone operator $H$, we propose a new algorithm, called the mirror-prox sliding met…
In this paper, we propose and study the iteration complexity of an inexact Douglas-Rachford splitting (DRS) method and a Douglas-Rachford-Tseng's forward-backward (F-B) splitting method for solving two-operator and four-operator monotone inclusions, respectively. The former method (although based on a slightly different mechanism of iteration) is motivated by the recent work of J. Eckstein and W. Yao, in which an inexact DRS method is derived from a special instance of the hybrid proximal extragradient (HPE) method of Solodov and Svaiter, while the latter one combines the proposed inexact DRS …
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.; …
A Variable-Metric Non-monotone Line Search Method for Mixed Variational Inequalities and Equilibrium Problems
King Fahd University of Petroleum & Minerals (KFUPM)（法赫德国王石油矿产大学）；Interdisciplinary Research Center (IRC) for Smart Mobility and Logistics, King Fahd University of Petroleum & Minerals (KFUPM)（法赫德国王石油矿产大学智能交通与物流交叉研究中心）
In this paper, a conceptual algorithm modifying the forward-backward-half-forward (FBHF) splitting method for solving three operator monotone inclusion problems is investigated. The FBHF splitting method adjusts and improves Tseng's forward-backward-forward (FBF) splitting method when the inclusion problem has a third-part operator that is cocoercive. The FBHF method recovers the FBF iteration (when this aforementioned part is zero), and it also works without using the widely used Lipschitz continuity assumption. The conceptual algorithm proposed in this paper also has those advantages, and it…
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 consider a class of N-player stochastic games of multi-dimensional singular control, in which each player faces a minimization problem of monotone-follower type with submodular costs. We call these games "monotone-follower games". In a not necessarily Markovian setting, we establish the existence of Nash equilibria. Moreover, we introduce a sequence of approximating games by restricting, for each natural number n, the players' admissible strategies to the set of Lipschitz processes with Lipschitz constant bounded by n. We prove that, for each n, there exists a Nash equilibrium of the approx…
In a previous work (Akian, Fodjo, 2016), we introduced a lower complexity probabilistic max-plus numerical method for solving fully nonlinear Hamilton-Jacobi-Bellman equations associated to diffusion control problems involving a finite set-valued (or switching) control and possibly a continuum-valued control. This method was based on the idempotent expansion properties obtained by McEneaney, Kaise and Han (2011) and on the numerical probabilistic method proposed by Fahim, Touzi and Warin (2011) for solving some fully nonlinear parabolic partial differential equations. A difficulty of the algor…
We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators a…
The proximal extrapolated gradient method \cite{Malitsky18a} is an extension of the projected reflected gradient method \cite{Malitsky15}. Both methods were proposed for solving the classic variational inequalities. In this paper, we investigate the projected reflected gradient method, in the general setting, for solving monotone inclusions involving Lipschitzian operators. As a result, we obtain a simple method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. We also show that one can improve the range of the stepsize of this method for the case …
University of Vienna（维也纳大学）；University of Southampton（南安普顿大学）
We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system which approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time di…
We consider the problem of solving dual monotone inclusions involving sums of composite parallel-sum type operators. A feature of this work is to exploit explicitly the cocoercivity of some of the operators appearing in the model. Several splitting algorithms recently proposed in the literature are recovered as special cases.
The aim of this survey is to present the main important techniques and tools from variational analysis used for first and second order dynamical systems of implicit type for solving monotone inclusions and non-smooth optimization problems. The differential equations are expressed by means of the resolvent (in case of a maximally monotone set valued operator) or the proximal operator for non-smooth functions. The asymptotic analysis of the trajectories generated relies on Lyapunov theory, where the appropriate energy functional plays a decisive role. While the most part of the paper is related …
We propose a splitting algorithm for solving a system of composite monotone inclusions formulated in the form of the extended set of solutions in real Hilbert spaces. The resluting algorithm is a an extension of the algorithm in [4]. The weak convergence of the algorithm proposed is proved. Applications to minimization problems is demonstrated.
In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partialΦ+B$, where $\partialΦ$ is the subdifferential of a convex lower semicontinuous function $Φ$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the converge…
