Strong Pseudomonotonicity Research Articles

A Golden Ratio Algorithm With Backward Inertial Step For Variational Inequalities

In this paper we study the convergence analysis of a Golden Ratio Algorithm with a backward inertial step and a fully adaptive step size procedure for the purpose of approximating solutions of variational inequalities in Hilbert spaces. We present a weak convergence result when the operator is quasi-monotone and locally Lipschitz continuous, and a strong convergence result in the setting of strong pseudo-monotonicity. Our algorithm features one operator evaluation and one projection computation at the current iteration. We recover interesting algorithms in the literature, for instance, the Golden Ratio Algorithm. Numerical experiments were conducted to validate the theoretical convergence analysis.

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined, and converges to the unique solution of the associated MVIP within a fixed time. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions.

Fast relaxed inertial Tseng’s method-based algorithm for solving variational inequality and fixed point problems in Hilbert spaces

Motivated and inspired by the works of Ceng et al. (2010) and Yao and Postolache (2012), we first study a relaxed inertial Tseng’s method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of fixed points of an κ-demicontractive mapping in real Hilbert spaces. The strong convergence of the algorithm is proved with conditions weaker than the conditions of other methods studied in the literature. Next, we also obtain an R-linear convergence rate of relaxed inertial Tseng’s method under strong pseudomonotonicity and Lipschitz continuity assumptions of the variational inequality mapping. As far as we know, these results have not been considered before in the literature. Finally, some numerical examples illustrate the effectiveness of our algorithms.

Fixed-time stability of projection neurodynamic network for solving pseudomonotone variational inequalities

In this paper, we propose a novel modified neurodynamic network (MNN) based on projection to solve pseudomonotone variational inequalities. Under the strong pseudo-monotonicity and Lipschitz continuous assumptions, the MNN admits a unique solution. The relations between the MNN and the corresponding neurodynamic network are obtained. Moreover, we establish the globally fixed-time stability of the MNN. The convergence time of the MNN is not depended on the initial conditions and uniformly bounded. Numerical examples are also reported to show the effectiveness and superiority of the MNN.

Fast alternated inertial projection algorithms for pseudo-monotone variational inequalities

In this paper, we propose an alternated inertial subgradient extragradient algorithm for variational inequalities with self-adaptive step-sizes and obtain weak and linear convergence results. We also obtain linear convergence results using an alternated inertial projected gradient algorithm for which knowledge of the modulus of strong pseudo-monotonicity and Lipschitz constant of the cost function are not needed. We compare numerically our algorithms with other projection-type algorithms in the literature.

Two fast converging inertial subgradient extragradient algorithms with variable stepsizes for solving pseudo-monotone VIPs in Hilbert spaces

In this work, we propose two new iterative schemes for finding an element of the set of solutions of a pseudo-monotone, Lipschitz continuous variational inequality problem in real Hilbert spaces. The weak and strong convergence theorems are presented. The advantage of the proposed algorithms is that they do not require prior knowledge of the Lipschitz constant of the variational inequality mapping and only compute one projection onto a feasible set per iteration as well as without using the sequentially weakly continuity of the associated mapping. Under additional strong pseudo-monotonicity and Lipschitz continuity assumptions, we obtain also an R-linear convergence rate of the proposed algorithm. Finally, some numerical examples are given to illustrate the effectiveness of the algorithms.

A new splitting algorithm for equilibrium problems and applications

"In this paper, we discuss a new splitting algorithm for solving equilibrium problems arising from Nash-Cournot oligopolistic equilibrium problems in electricity markets with non-convex cost functions. Under the strong pseudomonotonicity of the original bifunction and suitable conditions of the component bifunctions, we prove the strong convergence of the proposed algorithm. Our results improve and develop previously discussed extragradient-like splitting algorithms and general extragradient algorithms. We also present some numerical experiments and compare our algorithm with the existing ones."

"Extragradient method with a new adaptive step size for solving non-Lipschitzian pseudo-monotone variational inequalities"

"The purpose of this work is to develop a new version of the extragradient method for solving non-Lipschitzian and pseudo-monotone variational inequalities in real Hilbert spaces. First, we prove a sufficient condition for weak convergence of a proposed algorithm under relaxed assumptions. Next, under strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a Q-linear convergence rate of this algorithm. Our results improve some recent contributions in the literature on the extragradient method."

Existence results and two step proximal point algorithm for equilibrium problems on Hadamard manifolds

"In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic upper sign continuity of the equilibrium bifunction and under different kinds of coercivity conditions. We also study the existence of solutions of the equilibrium problems under properly quasimonotonicity of the equilibrium bifunction. We propose a two-step proximal point algorithm for solving equilibrium problems in the setting of Hadamard manifolds. The convergence of the proposed algorithm is studied under the strong pseudomonotonicity and Lipschitz-type condition. The results of this paper either extend or generalize several known results in the literature."

Strong convergence of inertial subgradient extragradient algorithm for solving pseudomonotone equilibrium problems

In this paper, we propose a new modified subgradient extragradient method for solving equilibrium problems involving pseudomonotone and Lipchitz-type bifunctions in Hilbert spaces. We establish the strong convergence of the proposed method under several suitable conditions. In addition, the linear convergence is obained under strong pseudomonotonicity assumption. Our results generalize and extend some related results in the literature. Finally, we provide numerical experiments to illustrate the performance of the proposed algorithm.

A new self-adaptive algorithm for solving pseudomonotone variational inequality problems in Hilbert spaces

ABSTRACT In this paper, we revisit the subgradient extragradient method for solving a pseudomonotone variational inequality problem with the Lipschitz condition in real Hilbert spaces. A new algorithm based on the subgradient extragradient method with the technique of choosing a new step size is proposed. The weak convergence of the proposed algorithm is established under the pseudomonotonicity and the Lipschitz continuity as well as without using the sequentially weakly continuity of the variational inequality mapping and the nonasymptotic convergence rate of the proposed algorithm is presented, while the strong convergence theorem of the proposed algorithm is also proved under the strong pseudomonotonicity and the Lipschitz continuity hypotheses. In order to show the computational effectiveness of our algorithm, some numerical results are provided.

Improved subgradient extragradient methods for solving pseudomonotone variational inequalities in Hilbert spaces

The purpose of this work to investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces. For solving this problem, we propose two new methods which combine advantages of the subgradient extragradient method and the projection contraction method. Similar to some recent developments, the proposed methods do not require the knowledge of the Lipschitz constant associated with the variational inequality mapping. Under suitable mild conditions, we establish the weak and strong convergence of the proposed algorithms. Moreover, linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions. Numerical examples in fractional programming and optimal control problems demonstrate the potential of our algorithms as well as compare their performances to several related results.

A Second Order Dynamical System and Its Discretization for Strongly Pseudo-monotone Variational Inequalities

We consider a second order dynamical system for solving variational inequalities in Hilbert spaces. Under standard conditions, we prove the existence and uniqueness of strong global solution of the proposed dynamical system. The exponential convergence of trajectories is established under strong pseudo-monotonicity and Lipschitz continuity assumptions. A discrete version of the proposed dynamical system leads to a relaxed inertial projection algorithm whose linear convergence is proved under suitable conditions on parameters. We discuss the possibility of extension to general monotone inclusion problems. Finally some numerical experiments are reported demonstrating the theoretical results.

A Dynamical System for Strongly Pseudo-monotone Equilibrium Problems

In this paper, we consider a dynamical system for solving equilibrium problems in the framework of Hilbert spaces. First, we prove that under strong pseudo-monotonicity and Lipschitz-type continuity assumptions, the dynamical system has a unique equilibrium solution, which is also globally exponentially stable. Then, we derive the linear rate of convergence of a discrete version of the proposed dynamical system to the unique solution of the problem. Global error bounds are also provided to estimate the distance between any trajectory and this unique solution. Some numerical experiments are reported to confirm the theoretical results.

Versions of the Subgradient Extragradient Method for Pseudomonotone Variational Inequalities

We develop versions of the subgradient extragradient method for variational inequalities in Hilbert spaces and establish sufficient conditions for their convergence. First we prove a sufficient condition for a weak convergence of a recent existing algorithm under relaxed assumptions. Then, we propose two other algorithms. Both weak and strong convergence of the considered algorithms are studied. Under additional strong pseudomonotonicity and Lipschitz continuity assumptions, we obtain also a $Q$ -linear convergence rate of these algorithms. Our results improve some recent contributions in the literature. Illustrative numerical experiments are also provided by the end of the paper.

The forward–backward–forward method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces

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 system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into Tseng’s forward–backward–forward algorithm with relaxation parameters, which we prove to converge also when it is applied to pseudo-monotone variational inequalities. In addition, we show that linear convergence is guaranteed under strong pseudo-monotonicity. Numerical experiments are carried out for pseudo-monotone variational inequalities over polyhedral sets and fractional programming problems.

An inertial modified algorithm for solving variational inequalities

The paper deals with an inertial-like algorithm for solving a class of variational inequality problems involving Lipschitz continuous and strongly pseudomonotone operators in Hilbert spaces. The presented algorithm can be considered a combination of the modified subgradient extragradient-like algorithm and inertial effects. This is intended to speed up the convergence properties of the algorithm. The main feature of the new algorithm is that it is done without the prior knowledge of the Lipschitz constant and the modulus of strong pseudomonotonicity of the cost operator. Several experiments are performed to illustrate the convergence and computational performance of the new algorithm in comparison with others having similar features. The numerical results have confirmed that the proposed algorithm has a competitive advantage over the existing methods.

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (i) We present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (ii) Under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of O(1/k) statements that were hitherto unavailable K in this regime. Notably, we optimize the initial steplength by obtaining an {\epsilon}-infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.

Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications

In this paper, we introduce two self-adaptive algorithms for solving a class of non-Lipschitz equilibrium problems. These algorithms are very simple in the sense that at each step, they require only one projection onto a feasible set. Their convergence can be established under quite mild assumptions. More precisely, the weak (strong) convergence of the first algorithm is proved under the pseudo-paramonotonicity (strong pseudomonotonicity) conditions, respectively. Especially, the convexity in the second argument of the involving bifunction is not required. In the second algorithm, the weak convergence is established under the pseudomonotonicity. Moreover, it is proved that under some additional conditions, the solvability of the equilibrium problem is equivalent to the boundedness of the sequences generated by the proposed algorithms. Some applications to the optimization problems and variational inequality problems as well as to transport equilibrium problems are also considered.

New inertial algorithm for a class of equilibrium problems

The article introduces a new algorithm for solving a class ofequilibrium problems involving strongly pseudomonotone bifunctions with Lipschitz-type condition. We describe how to incorporate the proximal-like regularized technique with inertial effects. The main novelty of the algorithm is that it can be done without previously knowing the information on the strongly pseudomonotone and Lipschitz-type constants of cost bifunction. A reasonable explain for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. Theorem of strong convergence is proved. In the case, when the information on the modulus of strong pseudomonotonicity and Lispchitz-type constant is known,the rate of linear convergence of the algorithm has been established. Several of numerical experiments are performed to illustrate the convergence of the algorithm and also compare it with other algorithms.