Search

In this work, we propose a new inertial method for solving strongly monotone variational inequality problems over the solution set of a split variational inequality and composed fixed point problem in real Hilbert spaces. Our method uses stepsizes that are generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the operator norm as well as the Lipschitz constant of the operator. In addition, we prove that the proposed method converges strongly to a minimum-norm solution of the problem without using the conventional two cases approach. Furthermore, we present some numerical experiments to show the efficiency and applicability of our method in comparison with other methods in the literature. The results obtained in this paper extend, generalize and improve results in this direction.

We study distributed algorithms for seeking a Nash equilibrium in a class of convex networked Nash 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 a gradient step to minimize her own cost function while sharing and retrieving information locally among her neighbors in the network. Using the reformulation of the Nash equilibrium problem based on the strong monotone augmented game mapping, we prove the convergence of this algorithm to a Nash equilibrium with a geometric rate. Furthermore, we introduce the Nesterov type acceleration for the gradient play algorithm. We demonstrate that, similarly to the accelerated algorithms in centralized optimization and variational inequality problems, our accelerated algorithm outperforms GRANE in the convergence rate. Moreover, to relax assumptions required to guarantee the strongly monotone augmented mapping, we analyze the restricted strongly monotone property of this mapping and prove geometric convergence of the distributed gradient play under milder assumptions.

It is the purpose of this paper to introduce an iterative process which converges strongly to a common point of the set of solutions of a finite family of generalized equilibrium problems, the set of fixed points of a finite family of continuous asymptotically quasi- $$\phi$$ -nonexpansive mappings in the intermediate sense, and the set of zeros of a finite family of $$\gamma$$ -inverse strongly monotone mappings in uniformly convex and uniformly smooth real Banach space. Our results improve and unify most of the results that have been proved for this important class of nonlinear mappings.

In this paper, an inverse-free dynamical system is built to solve the absolute value equations (AVEs), whose equilibrium points coincide with the solutions of the AVEs. Under proper assumptions, the equilibrium points of the dynamical system exist and could be (globally) asymptotically stable. In addition, with strongly monotone property, a global projection-type error bound is provided to estimate the distance between any trajectories and the unique equilibrium point. Compared with four existing dynamical systems for solving the AVEs, our method is inverse-free and is still valid even if 1 is an eigenvalue of the coefficient matrix. Some numerical simulations are given to show the effectiveness of the proposed method.

In this article, we propose centralized and distributed continuous-time penalty methods to find a Nash equilibrium for a generalized noncooperative game with shared inequality and equality constraints and private inequality constraints that depend on the player itself. By using the ℓ1 penalty function, we prove that the equilibrium of a differential inclusion is a normalized Nash equilibrium of the original generalized noncooperative game, and the centralized differential inclusion exponentially converges to the unique normalized Nash equilibrium of a strongly monotone game. Suppose that the players can communicate with their neighboring players only and the communication topology can be represented by a connected undirected graph. Based on a leader-following consensus scheme and singular perturbation techniques, we propose distributed algorithms by using the exact ℓ1 penalty function and the continuously differentiable squared ℓ2 penalty function, respectively. The squared ℓ2 penalty function method works for games with smooth constraints and the exact ℓ1 penalty function works for certain scenarios. The proposed two distributed algorithms converge to an η-neighborhood of the unique normalized Nash equilibrium and an -neighborhood of an approximated Nash equilibrium, respectively, with being a positive constant. For each 0 and each initial condition, there exists an such that for each 0, the convergence can be guaranteed where is a parameter in the algorithm. Ministry of Education (MOE) Nanyang Technological University This work was supported in part by the Singapore Ministry of Education Academic Research Fund Tier 1 RG180/17(2017-T1- 002-158) and in part by the Wallenberg-NTU Presidential Postdoctoral Fellow Set-Up Grant.

The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature.

In this paper, we propose an incremental constraint projection method (i.e., random or cyclic projection algorithm) for solving variational inequality problem with special structure, which the underlying mapping is strongly monotone and the constraint set is the intersection of a large number of simple closed convex sets. Compared with some existing projection type algorithms, the proposed method has two notable advantages: Its global convergence can be guaranteed without the Lischitz continuity of underlying mapping in almost sure sense; It just computes only one halfspace projection rather than the projection of the full or single constraint set at each iteration. Preliminary computational experience is also reported to illustrate the effectiveness of the proposed method.

In this paper we present sufficient conditions for convergence of projection and fixed-point algorithms used to compute dynamic user equilibrium with elastic travel demand (E-DUE). The assumption of strongly monotone increasing path delay operators is not needed. In its place, we assume path delay operators are merely weakly monotone increasing, a property assured by Lipschitz continuity, while inverse demand functions are strongly monotone decreasing. Lipschitz continuity of path delay is a very mild regularity condition. As such, nonmonotone delay operators may be weakly monotone increasing and satisfy our convergence criteria, provided inverse demand functions are strongly monotone decreasing. We illustrate convergence for nonmonotone path delays via a numerical example.

The purpose of this article is to study convergence analysis of quasi-variational inequalities using a projection-type method coupled with inertial extrapolation step. First, we give strong converg...

In this paper, we introduce a new and efficient algorithm for finding a common element of the set of common fixed points of a finite family of generalized demimetric mappings and the set of common ...

This article studies a generalized Nash equilibrium problem with coupling equality constraints and local action sets, where the cost function of each player has a general form that depends on the actions of other players in this game. In the case that the players cannot directly use the others’ actions, all players are allowed to estimate their opponents’ actions by communicating with their neighbors over a digraph. In this regard, continuous-time coordination dynamics are proposed for two kinds of directed communication topologies including weight-balanced and weight-unbalanced digraphs. When the pseudogradient is strongly monotone and Lipschitz continuous as well as the extended pseudogradient is Lipschitz continuous, it is theoretically shown that the proposed dynamics could solve the generalized Nash equilibrium problem with and without local action sets, respectively. Finally, the obtained theoretical results are illustrated by numerical simulations.

We revisit a unified methodology for the iterative solution of nonlinear equations in Hilbert spaces. Our key observation is that the general approach from [P. Heid and T. P. Wihler, Adaptive iterative linearization Galerkin methods for nonlinear problems, Math. Comp. 89 2020, 326, 2707–2734; P. Heid and T. P. Wihler, On the convergence of adaptive iterative linearized Galerkin methods, Calcolo 57 2020, Paper No. 24] satisfies an energy contraction property in the context of (abstract) strongly monotone problems. This property, in turn, is the crucial ingredient in the recent convergence analysis in [G. Gantner, A. Haberl, D. Praetorius and S. Schimanko, Rate optimality of adaptive finite element methods with respect to the overall computational costs, preprint 2020]. In particular, we deduce that adaptive iterative linearized finite element methods (AILFEMs) lead to full linear convergence with optimal algebraic rates with respect to the degrees of freedom as well as the total computational time.

In this work we introduce and analyse a new multiscale method for strongly nonlinear monotone equations in the spirit of the localized orthogonal decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems, which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors. The results neither require structural assumptions on the coefficient such as periodicity or scale separation nor higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including the stationary Richards equation confirm the theory and underline the applicability of the method.

We consider a C 1 , α smooth flow in R d which is “strongly monotone” with respect to a cone C of rank k, a closed set that contains a linear subspace of dimension k and no linear subspaces of higher dimension. We prove that orbits with initial data from an open and dense subset of the phase space are either pseudo-ordered or convergent to equilibria. This covers the celebrated Hirsch's Generic Convergence Theorem in the case k = 1 , yields a generic Poincare-Bendixson Theorem for the case k = 2 , and holds true with arbitrary dimension k. Our approach involves the ergodic argument using the k-exponential separation and the associated k-Lyapunov exponent (that reduces to the first Lyapunov exponent if k = 1 ).

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly monotone and Lipschitz-type assumptions of the bifunction. By combining with parallel techniques, the convergence result is also established for the equilibrium problem involving a finite system of demicontractive mappings. Several fundamental experiments are provided to illustrate the numerical behavior of the proposed algorithm and comparison with other known algorithms is studied.

In this paper, strong convergence results for $ \alpha - $inverse strongly monotone operators under new algorithms in the framework of Hilbert spaces are discussed. Our algorithms are the combination of the inertial Mann forward-backward method with the CQ-shrinking projection method and viscosity algorithm. Our methods lead to an acceleration of modified inertial Mann Halpern and viscosity algorithms. Later on, numerical examples to illustrate the applications, performance, and effectiveness of our algorithms are presented.

In the paper, we consider bilevel problems: variational inequality problems over the set of solutions of the equilibrium problem. Finding normal Nash equilibrium is an example of such a problem. To solve these problems, an iterative algorithm is proposed that combines the ideas of the two-stage proximal method, adaptability, and iterative regularization. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of information on bifunction’s Lipschitz constants and operator’s Lipschitz and strong monotonicity constants. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators, the theorem on strong convergence of sequences generated by the algorithm is proved. The proposed algorithm is shown to be applicable to monotone bilevel variational inequalities in Hilbert spaces.

For C 1 -smooth strongly monotone discrete-time dynamical systems, we prove dynamics alternatives, which concludes that any compact orbit is either asymptotic to a linearly stable cycle; or manifestly unstable. For this purpose we improve several properties of the exponential separation for continuous maps. The generic convergence to cycles is obtained as a by-product of the dynamics alternatives.

In this paper, we propose modified Tseng's extragradient methods with self-adaptive step size for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mappin...

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of a split variational inequality and fixed point problem. Strong convergence of the iterative process is proved. In particular, the problem of finding a common solution to a variational inequality with pseudomonotone mapping and a fixed point problem involving demicontractive mapping is also studied. Besides, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each step. A simple numerical example is given to illustrate the proposed algorithm.

Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

The article introduces a new algorithm for solving a class of variational inequality problems for monotone operators and system of nonlinear variational inequalities problems for two inverse strongly monotone operators. We describe how to incorporate the extragradient like technique based on altering points technique with inertial effects. A weak convergence theorem is established for the proposed algorithm. Numerical examples are performed to illustrate the numerical efficiency of the algorithm and compare with other algorithms.

In this article, we consider the problem of finding zeros of monotone inclusions of three operators in real Hilbert spaces, where the first operator’s inverse is strongly monotone and the third is linearly composed, and we suggest an extended splitting method. This method allows relative errors and is capable of decoupling the third operator from linear composition operator well. At each iteration, the first operator can be processed with just a single forward step, and the other two need individual computations of the resolvents. If the first operator vanishes and linear composition operator is the identity one, then it coincides with a known method. Under the weakest possible conditions, we prove its weak convergence of the generated primal sequence of the iterates by developing a more self-contained and less convoluted techniques. Our suggested method contains one parameter. When it is taken to be either zero or two, our suggested method has interesting relations to existing methods. Furthermore, we did numerical experiments to confirm its efficiency and robustness, compared with other state-of-the-art methods.

We study the single projection algorithm of Tseng for solving a variational inequality problem in a 2-uniformly convex Banach space. The underline cost function of the variational inequality is assumed to be monotone and Lipschitz continuous. A weak convergence result is obtained under reasonable assumptions on the variable step-sizes. We also give the strong convergence result for when the underline cost function is strongly monotone and Lipchitz continuous. For this strong convergence case, the proposed method does not require prior knowledge of the modulus of strong monotonicity and the Lipschitz constant of the cost function as input parameters, rather, the variable step-sizes are diminishing and non-summable. The asymptotic estimate of the convergence rate for the strong convergence case is also given. For completeness, we give another strong convergence result using the idea of Halpern’s iteration when the cost function is monotone and Lipschitz continuous and the variable step-sizes are bounded by the inverse of the Lipschitz constant of the cost function. Finally, we give an example of a contact problem where our proposed method can be applied.

In this paper, by basing on the inexact subgradient and projection methods presented by Santos et al. (Comput. Appl. Math. 30: 91–107, 2011), we develop subgradient projection methods for solving strongly monotone equilibrium problems with pseudomonotone equilibrium constraints. The problem usually is called monotone bilevel equilibrium problems. We show that this problem can be solved by a simple and explicit subgradient method. The strong convergence for the proposed algorithms to the solution is guaranteed under certain assumptions in a real Hilbert space. Numerical illustrations are given to demonstrate the performances of the algorithms.

This paper addresses an optimal impulsive control problem whose trajectories are functions of bounded variation and impulsive controls are regular vector measures. This problem is characterized by two main features. First, the dynamical control system to be considered may not possess the so-called well-posedness property. Second, the constraints on the one-sided limits of states are presented. Such constraints are interpreted as multipoint state constraints. For this problem, we derive global optimality conditions based on using of compound Lyapunov type functions which possess strongly monotone properties with respect to the control system. As a motivating case, a model of advertising expenses optimization for mutually complementary products is considered. For this model, we propose four variants of resolving sets of Lyapunov type functions and explain the technique of applying the optimality conditions.

LetEbe a real Banach space with dual space$E^{*}$E∗. A new class ofrelatively weakJ-nonexpansive maps,$T:E\rightarrow E^{*}$T:E→E∗, is introduced and studied. An algorithm to approximate a common element ofJ-fixed points for a countable family of relatively weakJ-nonexpansive maps and zeros of a countable family of inverse strongly monotone maps in a 2-uniformly convex and uniformly smooth real Banach space is constructed. Furthermore, assuming existence, the sequence of the algorithm is proved to converge strongly. Finally, a numerical example is given to illustrate the convergence of the sequence generated by the algorithm.

We provide a distributed algorithm to learn a Nash equilibrium in a class of non-cooperative games with strongly monotone mappings and unconstrained action sets. Each player has access to her own smooth local cost function and can communicate to her neighbors in some undirected graph. We consider a distributed communication-based gradient algorithm. For this procedure, we prove geometric convergence to a Nash equilibrium. In contrast to our previous works Tatarenko et al. (2018); Tatarenko et al. (2019), where the proposed algorithms required two parameters to be set up and the analysis was based on a so called augmented game mapping, the procedure in this work corresponds to a standard distributed gradient play and, thus, only one constant step size parameter needs to be chosen appropriately to guarantee fast convergence to a game solution. Moreover, we provide a rigorous comparison between the convergence rate of the proposed distributed gradient play and the rate of the GRANE algorithm presented in Tatarenko et al. (2019). It allows us to demonstrate that the distributed gradient play outperforms the GRANE in terms of convergence speed.

In this article, we design a projection type iterative algorithm with two inertial steps for solving quasi-variational inequalities with Lipschitz continuous and strongly monotone mappings in real Hilbert spaces. We establish different strong convergence results through this algorithm. We give a non-trivial example to validate one of our results and to illustrate the efficiency of the proposed algorithm compared with an already existing one. We also present some numerical experiments to demonstrate the potential applicability and computing performance of our algorithm compared with some other algorithms existing in the literature. The results obtained herein are generalizations and substantial improvements of some earlier results.

In this paper, we study a class of derivative-free nonmonotone line search methods for solving nonlinear systems of equations, which includes the method N-DF-SANE proposed in Cheng and Li (IMA J Numer Anal 29:814–825, 2009). These methods correspond to derivative-free optimization methods applied to the minimization of a suitable merit function. Assuming that the mapping defining the system of nonlinear equations has Lipschitz continuous Jacobian, we show that the methods in the referred class need at most $${\mathcal {O}}\left( |\log (\epsilon )|\epsilon ^{-2}\right) $$ function evaluations to generate an $$\epsilon $$ -approximate stationary point to the merit function. For the case in which the mapping is strongly monotone, we present two methods with evaluation-complexity of $${\mathcal {O}}\left( |\log (\epsilon )|\right) $$ .

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

In this paper, we introduce an inertial projection-type method with different updating strategies for solving quasi-variational inequalities with strongly monotone and Lipschitz continuous operators in real Hilbert spaces. Under standard assumptions, we establish different strong convergence results for the proposed algorithm. Primary numerical experiments demonstrate the potential applicability of our scheme compared with some related methods in the literature.

This paper concerns with a new three-operator splitting algorithm for solving a class of variational inclusions. The main advantage of the proposed algorithm is that it can be easily implemented without the prior knowledge of Lipschitz constant, strongly monotone constant and cocoercive constant of component operators. A reason explained for this is that the algorithm uses a sequence of stepsizes which is diminishing and non-summable. The strong convergence of the algorithm is established. Several fundamental numerical experiments are given to illustrate the behavior of the new algorithm and compare it with other algorithms.

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.

In the present paper we prove some uniqueness results for weak solutions to a class of problems, whose prototype is $$\begin{cases}-\rm{div} & ((\varepsilon+|\triangledown{u}|^2)\frac{p-2}{2}\triangledown{u}\varphi)=f\varphi\;\;\;\;\;\rm{in}\;\;\Omega\\u=0 &\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \rm{on}\;\partial\Omega,\end{cases}$$ where e ≥ 0, 1 1) with Gauss measure less than one and datum f belongs to the natural dual space. When p ≤ 2 we obtain a uniqueness result for e = 0, while for p > 2 we have to consider e > 0 unless the sign of f is constant. Some counterexamples are given too.

Let X be a uniformly convex and uniformly smooth real Banach space with dual space $X^{*}$ . In this paper, a Mann-type iterative algorithm that approximates the zero of a generalized-Φ-strongly monotone map is constructed. A strong convergence theorem for a sequence generated by the algorithm is proved. Furthermore, the theorem is applied to approximate the solution of a convex optimization problem, a Hammerstein integral equation, and a variational inequality problem. This theorem generalizes, improves, and complements some recent results. Finally, examples of generalized-Φ-strongly monotone maps are constructed and numerical experiments which illustrate the convergence of the sequence generated by our algorithm are presented.