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In this paper, we present an inertial projection and contraction algorithm (IPCA for short) for variational inequality problems (VIP for short) without Lipschitz continuity of the underlying mapping in Euclidean space. To obtain a larger step, the next iterate point of IPCA is generated by projecting the current iterate point onto a selected half-space, which is selected from finite half-spaces (one half-space can strictly separate the current iterate point from the solution set of VIP and other half-spaces are all contain the solution set of VIP) and has the largest distance to the current iterate point. Moreover, a new initial step-size strategy is used to accelerate IPCA. The global convergence of the sequence generated by IPCA is established. Numerical experiments show that IPCA can accelerate PCA both from the number of iterations point of view and the number of projections point of view. [ABSTRACT FROM AUTHOR]

In this paper, we present an inertial Popov extragradient projection algorithm for solving multi-valued variational inequality problems in finite-dimensional Euclidean space. The global convergence of this algorithm is proved whenever the underlying mapping is Lipschitz continuous and pseudomonotone on the feasible set. Numerical experiments show that new algorithm is more efficient than algorithm of Ye [An improved projection method for solving generalized variational inequality problems. Optimization. 2018;67:1–11] whenever the underlying mapping is Lipschitz continuous. Here inertial technique can accelerate extragradient algorithm although the underlying mapping is multi-valued. [ABSTRACT FROM AUTHOR]

In a very interesting paper (SIAM J. Control Optim. 37(3): 765–776, 1999), Solodov and Svaiter introduced an effective projection algorithm with linesearch for finding a solution of a variational inequality problem (VIP) in Euclidean space. They showed that the iterative sequence generated by their algorithm converges to a solution of (VIP) under the main assumption that the cost mapping is pseudomonotone and continuous. In this paper, we propose to modify this algorithm for solving variational inequality problems in which the cost mapping is not required to be satisfied any pseudomonotonicity. Moreover, we do not use the embedded projection methods as in methods used in literature and the linesearch procedure is not necessary when the cost mapping is Lipschitz. Several numerical examples are also provided to illustrate the efficient of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The paper aims to introduce inverse tensor variational inequalities and analyze their application to an economic control equilibrium model. More precisely, some existence and uniqueness results are established and the well-posedness analysis is investigated. Moreover, the Tikhonov regularization method is extended to tensor inverse problems to study them when they are ill-posed. Lastly, the policymaker's point of view for the oligopolistic market equilibrium problem is introduced. The equivalence between the equilibrium conditions and a suitable inverse tensor variational inequality is established. [ABSTRACT FROM AUTHOR]

In this paper, we define a generalized resolvent operator involving a 풢 ⁢ (⋅ , ⋅) -co-monotone mapping for solving a generalized variational inclusion problem. We show that the resolvent operator under consideration is single-valued as well as Lipschitz continuous. An iterative algorithm is constructed to obtain the approximate solution of our problem. An example is provided to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the topological structure of solution sets of monotone vector variational inequalities (VVIs). We show that if the weak Pareto solution set of a monotone VVI is disconnected, then each connected component of the set is unbounded. Similarly, this property holds for the proper Pareto solution set. Two open questions on the topological structure of the solution sets of (symmetric) monotone VVIs are raised at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of the split feasibility problem with multiple output sets in real Hilbert spaces. The strong convergence of the proposed algorithm is proved without knowing any information of the Lipschitz and strongly monotone constants of the mapping. In addition, the implementation of the algorithm does not require the computation or estimation of the norms of the given bounded linear operators. Special cases are considered. Finally, a numerical experiment has been carried out to illustrate the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we primary establish Hölder and Lipschitz continuity of solutions to abstract dynamical mixed equilibrium problems, DMEP for short, which we apply to obtain quantitative stability for parametric differential variational inclusions. Then, by involving key conditions on Fitzpatrick transform of equilibrium bifunctions introduced and studied in Chbani et al. [From convergence of dynamical equilibrium systems to bilevel hierarchical Ky Fan minimax inequalities and applications. J Minimax Theory Appl. 2019;4:231–270], we derive further quantitative stability for a parametric bilevel equilibrium problem which is regarded in our approach as a limit problem of the dynamic model DMEP. The obtained abstract result on bilevel equilibria is thereby applied to parametric mathematical programs with equilibrium constraints. A specific applied model illustrating the meaning of the involved parameters is also discussed with respect to optimal control problems whose state equation is defined by a variational inequality. We also report a numerical illustration to highlight the convergence and estimation rates we obtained and support our theoretical results. [ABSTRACT FROM AUTHOR]

It is well known that the monotonicity of the underlying mapping of variational inequalities plays a central role in the convergence analysis. In this paper, we propose an infeasible projection algorithm (IPA for short) for nonmonotone variational inequalities. The next iteration point of IPA is generated by projecting a vector onto a half-space. Hence, the computational cost of computing the next iteration point of IPA is much less than the algorithm of Ye and He (Comput. Optim. Appl. 60, 141–150, 2015) (YH for short). Moreover, if the underlying mapping is Lipschitz continuous with its modulus is known, by taking suitable parameters, IPA requires only one projection onto the feasible set per iteration. The global convergence of IPA is obtained when the solution set of its dual variational inequalities is nonempty. Moreover, if in addition error bound holds, the convergence rate of IPA is Q-linear. IPA can be used for a class of quasimonotone variational inequality problems and a class of quasiconvex minimization problems. Comparing with YH and Algorithm 2 in Deng, Hu and Fang (Numer. Algor. 86, 191–221, 2021) (DHF for short) by solving high-dimensional nonmonotone variational inequalities, numerical experiments show that IPA is much more efficient than YH and DHF from CPU time point of view. Moreover, IPA is less dependent on the initial value than YH and DHF. [ABSTRACT FROM AUTHOR]

In the era of highly dynamicity and uncertainty, it is of great importance to analyze the strategic decision process for companies’ technology acquisition between internal and external. In particular, the Internet of Things (IoT) has emerged as one of the crucial technologies that contribute to the resiliency and flexibility of companies willing to remain competitive in the markets. Despite the perceived advantages of IoT, it remains unclear what drive companies to adopt IoT in their businesses, due to strategic considerations about technology adoption and strategy selection. In that regard, this paper considers the relative payoffs among technology manufacturers, firms and demand markets, using game theory models and numerical analysis. Precisely by adopting a supply chain network perspective, our game theoretic representation assumes that technology manufacturers produce a technology that sell to firms, which in turn use the acquired technology to produce a commodity that is designed to be sold in the demand markets. Assuming that all the agents seek to maximize their profit functions, whereas the consumers want to satisfy their commodity requirement, in making their consumption decisions, the network structure of the problem is identified, the equilibrium conditions are derived, and the finite-dimensional variational inequality formulation is established. Finally, making use of the variational inequality theory, an existence and uniqueness theorem for the equilibrium distribution is obtained. The findings provide implications for competitive enterprises evaluating whether to adopt a new technology, identifying what conditions lead to win-win outcomes. In particular, firms must carefully consider the transaction costs both with technology manufacturers and demand markets, as key drivers of strategic considerations about the source of innovation. More precisely, when confronted with limited resources and the speed at which technology develop in the market, our findings suggest to companies to pursue a buy strategy, as it allows to gain time and benefit from lower innovation costs with respect to the alternative make strategy. [ABSTRACT FROM AUTHOR]

This paper settles an open and challenging question pertaining to the design of simple and optimal high-order methods for solving smooth and monotone variational inequalities (VIs). A VI involves finding x⋆∈X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^\star \in {\mathcal {X}}$$\end{document} such that ⟨F(x),x-x⋆⟩≥0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\langle F(x), x - x^\star \rangle \ge 0$$\end{document} for all x∈X\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x \in {\mathcal {X}}$$\end{document}. We consider the setting in which F:Rd→Rd\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F: {\mathbb {R}}^d \rightarrow {\mathbb {R}}^d$$\end{document} is smooth with up to (p-1)th\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(p-1)^{\text {th}}$$\end{document}-order derivatives. For p=2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p = 2$$\end{document}, the cubic regularization of Newton’s method has been extended to VIs with a global rate of O(ϵ-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\epsilon ^{-1})$$\end{document} (Nesterov in Cubic regularization of Newton’s method for convex problems with constraints, Tech. rep., Université catholique de Louvain, Center for Operations Research and Econometrics (CORE), 2006). An improved rate of O(ϵ-2/3loglog(1/ϵ))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\epsilon ^{-2/3}\log \log (1/\epsilon ))$$\end{document} can be obtained via an alternative second-order method, but this method requires a nontrivial line-search procedure as an inner loop. Similarly, the existing high-order methods based on line-search procedures have been shown to achieve a rate of O(ϵ-2/(p+1)loglog(1/ϵ))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\epsilon ^{-2/(p+1)}\log \log (1/\epsilon ))$$\end{document} (Bullins and Lai in SIAM J Optim 32(3):2208–2229, 2022; Jiang and Mokhtari in Generalized optimistic methods for convex–concave saddle point problems, 2022; Lin and Jordan in Math Oper Res 48(4):2353–2382, 2023). As emphasized by Nesterov (Lectures on convex optimization, vol 137, Springer, Berlin, 2018), however, such procedures do not necessarily imply the practical applicability in large-scale applications, and it is desirable to complement these results with a simple high-order VI method that retains the optimality of the more complex methods. We propose a pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order method that does not require any line search procedure and provably converges to a weak solution at a rate of O(ϵ-2/(p+1))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\epsilon ^{-2/(p+1)})$$\end{document}. We prove that our pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order method is optimal in the monotone setting by establishing a lower bound of Ω(ϵ-2/(p+1))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega (\epsilon ^{-2/(p+1)})$$\end{document} under a generalized linear span assumption. A restarted version of our pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order method attains a linear rate for smooth and pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order uniformly monotone VIs and another restarted version of our pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order method attains a local superlinear rate for smooth and strongly monotone VIs. Further, the similar pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order method achieves a global rate of O(ϵ-2/p)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$O(\epsilon ^{-2/p})$$\end{document} for solving smooth and nonmonotone VIs satisfying the Minty condition. Two restarted versions attain a global linear rate under additional pth\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p^{\text {th}}$$\end{document}-order uniform Minty condition and a local superlinear rate under additional strong Minty condition. [ABSTRACT FROM AUTHOR]

In this paper, a new class of problem known as the generalized bilevel mixed equilibrium problem with perturbations (GBMEPP) has been introduced on a real Banach space. Based on the diameter and the measure of non-compactness of the approximate solution set of the above stated problem, criteria and characterizations have been derived for Levitin–Polyak (LP) well-posedness. A sufficient condition for the LP well-posedness has also been derived assuming the boundedness of the approximate solution set. [ABSTRACT FROM AUTHOR]

In this paper, we construct an iterative algorithm for solving a multitime-type variational inequality problem and prove its convergence. Moreover, we study the weak sharp solutions for the multitime-type variational inequality problem and investigate the finite termination property of the constructed iterative algorithm. [ABSTRACT FROM AUTHOR]

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates. [ABSTRACT FROM AUTHOR]

In this paper, the weak sharpness property of solution sets of a split variational inequality problem is studied. Further, finite convergence analysis of the iterative methods for solving the split variational inequality problem and its special case, split feasibility problem are investigated. We also provide an application of split variational inequality problem in traffic network analysis and numerically illustrate our derived results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about monotone vector fields, which is a typical extension to the theory of convex functions, by exploiting the tangent space structure. This new approach to monotonicity in CAT (0) spaces stands in opposed to the monotonicity defined earlier in CAT (0) spaces by Khatibzadeh and Ranjbar [J. Aust. Math. Soc. 103(1), 70–90 (2017).] and Chaipunya and Kumam [Optimization 66(10), 1647–1665 (2017).]. In particular, this new concept extends the theory from both Hilbert spaces and Hadamard manifolds, while the known concept barely has any obvious relationship to the theory in Hadamard manifolds. We also study the corresponding resolvents and Yosida approximations of a given monotone vector field and derive many of their important properties. Finally, we prove a generation theorem by showing convergence of an exponential formula applied to resolvents of a monotone vector field. Our findings improve several known results in the literature including generation theorems of Jost [AMS/IP Stud. Adv. Math., vol. 8, pp. 1–47. Amer. Math. Soc., Providence, RI (1998)], Mayer [Comm. Anal. Geom. 6(2), 199–253 (1998).], Stojkovic [Adv. Calc. Var. 5(1), 77–126 (2012).], and Bačák [Convex analysis and optimization in Hadamard spaces, De Gruyter Series in Nonlinear Analysis and Applications, vol. 22. De Gruyter, Berlin (2014).] for proper, convex, lower semicontinuous functions in the context of complete CAT (0) spaces, and also by Iwamiya and Okochi [Nonlinear Anal. 54(2), 205–214 (2003).] for monotone vector fields in the context of Hadamard manifolds. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of the Galerkin method and regularization for variational inequalities with pseudomonotone operators in the sense of Brézis. Namely, we prove that under certain conditions, the solutions of the Galerkin equations and regularized variational inequalities converge strongly to a solution of the original variational inequality in reflexive Banach spaces. An application for obstacle problems is given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study an optimal control problem governed by mixed equilibrium problems described by the sum of a maximal monotone bifunction and a bifunction which is pseudomonotone in the sense of Brézis/quasimonotone. Our motivation comes from the fact that many control problems, whose state system is a variational inequality problem or a nonlinear evolution equation or a hemivariational inequality problem, can be formulated as a control problem governed by a mixed equilibrium problem. There are different techniques to study optimal control problems governed by nonlinear evolution equations, variational inequalities or hemivariational inequalities in the literature. However, our technique is completely different from existing ones. It is based on the Mosco convergence and recent results in the theory of equilibrium problems. As an application, we study optimal control problems governed by elliptic variational inequalities with additional state constraints. [ABSTRACT FROM AUTHOR]

In this paper, a computational technique based on the pseudo-spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo-spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well-developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a system of generalized mixed vector equilibrium problems involving set-valued variational-like inequalities (SGMVEP, in short) in the general setting of Banach spaces. First, we construct an associated vector auxiliary problem and establish existence results for it. This leads to the generation of an iterative algorithm for solving SGMVEP. Then, under weaker assumptions, the strong convergence of the proposed algorithm is established. The method adopted here is based on the vector auxiliary principle and the obtained results improve some existing results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parallel descent LQP alternating direction method for solving structured variational inequality with three separable operators. The $O(1/t)$ convergence rate for this method is studied. We also present some numerical examples to illustrate the efficiency of the proposed method. The results presented in this paper extend and improve some well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we establish explicit upper bounds for the number of connected components in the proper Pareto solution sets and the weak Pareto solution sets of polynomial vector variational inequalities. Consequently, upper bounds for the numbers of connected components in the stationary point sets, the proper stationary point sets, and the weak Pareto solution sets of polynomial vector optimization problems are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variational-type inequality and study its weak sharp solutions in terms of a dual gap function, under certain assumptions and convex notion of a functional. Moreover, we aim to constitute the relationship between the minimum principle sufficiency property and weak sharp solutions of the considered variational-type inequality. A numerical result is also constructed to give a better insight for the main derived result. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the introduction a new class of equilibrium problems named generalized multivalued equilibrium-like problems which includes the classes of hemiequilibrium problems, equilibrium-like problems, equilibrium problems, hemivariational inequalities, and variational inequalities as special cases. By utilizing the auxiliary principle technique, some new predictor-corrector iterative algorithms for solving them are suggested and analyzed. The convergence analysis of the proposed iterative methods requires either partially relaxed monotonicity or jointly pseudomonotonicity of the bifunctions involved in generalized multivalued equilibrium-like problem. Results obtained in this paper include several new and known results as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak sharp solutions for nonsmooth variational inequalities and give a characterization in terms of error bound. Some characterizations of solution set of nonsmooth variational inequalities are presented. Under certain conditions, we prove that the sequence generated by an algorithm for finding a solution of nonsmooth variational inequalities terminates after a finite number of iterates provided that the solutions set of a nonsmooth variational inequality is weakly sharp. We also study the finite termination property of the gradient projection method for solving nonsmooth variational inequalities under weak sharpness of the solution set. [ABSTRACT FROM AUTHOR]

Most mathematical models arising in stationary filtration processes as well as in the theory of soft shells can be described by single-valued or generalized multivalued pseudomonotone mixed variational inequalities with proper convex nondifferentiable functionals. Therefore, for finding the minimum norm solution of such inequalities, the current paper attempts to introduce a modified two-layer iteration via a boundary point approach and to prove its strong convergence. The results here improve and extend the corresponding recent results announced by Badriev, Zadvornov and Saddeek (Differ. Equ. 37:934-942, 2001). [ABSTRACT FROM AUTHOR]

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers. [ABSTRACT FROM AUTHOR]

This paper studies the local analysis of equations on a product U ×  U of Banach spaces, whose variables lie in a subset having the special property that it is locally Lipschitz-homeomorphic to an open subset of U. A prominent example, to which we devote most of the paper, is a system of equations whose variables lie in the graph of a maximal monotone operator. This general formulation covers many specific problems of interest, and our objective is to understand the local behavior of solutions of such equations when they depend on parameters. We analyze this local behavior in stages, the first stage being to apply a tailored implicit-function theorem to an abstract formulation of the problem, thereby producing a set of results applicable to any particular problem instance. The second stage is to specialize the analysis to a Hilbert space H, with the subset mentioned above being the graph of a maximal monotone operator on H. This makes the results of the first stage applicable to many variational problems of practical importance. We then develop in detail the analytical steps to apply these results to finite-dimensional variational conditions with constraints of generalized nonlinear-programming type. The conditions thus identified generalize the strong second-order sufficient condition and linear-independence constraint qualification of nonlinear programming. A detailed example brings out some of the issues involved in practical implementation of this method. It also shows that aspects of representation (problem formulation) can strongly influence the feasibility of local analysis. This sensitivity to representation does not seem to be well known. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new incremental constraint projection algorithm for solving variational inequalities, where the underlying function is monotone plus and Lipschitz continuous. The algorithm consists two steps. In the first step, we compute a predictor point. This procedure requires a single random projection onto some setand employs an Armijo-type linesearch along a feasible direction. Then in the second step an iterate is obtained as the random projection of some point onto the setwhich we have used in the first step. The incremental constraint projection algorithm is considered for random selection and for cyclic selection of the samples. Accordingly, this algorithm is named random projection algorithm and cyclic projection algorithm. The method is shown to be globally convergent to a solution of the variational inequality problem in almost sure sense both random projection method and cyclic projection method. We provide some computational experiments and compare the efficiency of random projection method and cyclic projection method with some known algorithms. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to introduce and study bilevel vector equilibrium problems. We first establish some existence results for solutions of vector equilibrium problems and mixed vector equilibrium problems. We study the existence of solutions of bilevel vector equilibrium problems by considering a vector Thikhonov-type regularization procedure. By using this regularization procedure and existence results for mixed vector equilibrium problems, we establish some existence results for solutions of bilevel vector equilibrium problems. By using the auxiliary principle, we propose an algorithm for finding the approximate solutions of bilevel vector equilibrium problems. The strong convergence of the proposed algorithm is also studied. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of bilevel variational inequalities with hierarchical nesting structure. We first of all get the existence of a solution for this problem by using the Himmelberg fixed point theorem. Then the uniqueness of the solution for an upper-level variational inequality is given under some mild conditions. By using gap functions of the upper-level and lower-level variational inequalities, we transform bilevel variational inequalities into a one-level variational inequality. Moreover, we propose two iterative algorithms to find the solutions of the bilevel variational inequalities. Finally, the convergence of the proposed algorithm is derived under some mild conditions. [ABSTRACT FROM AUTHOR]

In this paper, we consider two classes of generalized mixed vector equilibrium problems and mixed vector equilibrium problems, and propose some gap functions by using a new method, which is different from the previously known methods used in the literature. Finally, error bounds are obtained for the underlying mixed vector equilibrium problems in terms of regularized gap functions without using projection method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variational inequality problem (VIP) defined by a maximal monotone operator and a feasible set defined by convex inequality constraints and bounds on the variables. A Lagrangian dual method for solving this problem is presented and its convergence is proved. [ABSTRACT FROM PUBLISHER]

In this paper, we introduce a modified Krasnoselski–Mann type iterative method for capturing a common solution of a split mixed equilibrium problem and a hierarchical fixed point problem of a finite collection of k-strictly pseudocontractive nonself-mappings. Many of the algorithms for solving the split mixed equilibrium problem involve a step size which depends on the norm of a bounded linear operator. Since the computation of the operator norm is very difficult, we formulate our iterative algorithm in such a way that the implementation of the proposed algorithm does not require any prior knowledge of operator norm. Weak convergence results are established under mild conditions. We also establish strong convergence results for a certain class of hierarchical fixed point and split equilibrium problem. Our results generalize some important results in the recent literature. [ABSTRACT FROM AUTHOR]