The projection and contraction methods for finding common solutions to variational inequality problems

In this paper, we firstly introduce two projection and contraction methods for finding common solutions to variational inequality problems involving monotone and Lipschitz continuous operators in Hilbert spaces. Then, by modifying the two methods, we propose two hybrid projection and contraction methods. Both weak and strong convergence are investigated under standard assumptions imposed on the operators. Also, we generalize some methods to show the existence of solutions for a system of generalized equilibrium problems. Finally, some preliminary experiments are presented to illustrate the advantage of the proposed methods.     

Accelerated hybrid and shrinking projection methods for variational inequality problems

ABSTRACT

In this paper, we introduce several new extragradient-like approximation methods for solving variational inequalities in Hilbert spaces. Our algorithms are based on Tseng's extragradient method, subgradient extragradient method, inertial method, hybrid projection method and shrinking projection method. Strong convergence theorems are established under appropriate conditions. Our results extend and improve some related results in the literature. In addition, the efficiency of our algorithms is shown through numerical examples which are defined by the hybrid projection methods.     

Two-step projection methods for a system of variational inequality problems in Banach spaces

Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E and let Π _C be a sunny nonexpansive retraction from E onto C. Let the mappings ${T, S: C \to E}$ be γ ₁-strongly accretive, μ ₁-Lipschitz continuous and γ ₂-strongly accretive, μ ₂-Lipschitz continuous, respectively. For arbitrarily chosen initial point ${x^0 \in C}$ , compute the sequences {x ^k } and {y ^k } such that ${\begin{array}{ll} \quad y^k = \Pi_C[x^k-\eta S(x^k)],\ x^{k+1} = (1-\alpha^k)x^k+\alpha^k\Pi_C[y^k-\rho T(y^k)],\quad k\geq 0, \end{array}}$ where {α ^k } is a sequence in [0,1] and ρ, η are two positive constants. Under some mild conditions, we prove that the sequences {x ^k } and {y ^k } converge to x* and y*, respectively, where (x*, y*) is a solution of the following system of variational inequality problems in Banach spaces: ${\left\{\begin{array}{l}\langle \rho T(y^*)+x^*-y^*,j(x-x^*)\rangle\geq 0, \quad\forall x \in C,\\langle \eta S(x^*)+y^*-x^*,j(x-y^*)\rangle\geq 0,\quad\forall x \in C.\end{array}\right.}$ Our results extend the main results in Verma (Appl Math Lett 18:1286–1292, 2005) from Hilbert spaces to Banach spaces. We also obtain some corollaries which include some results in the literature as special cases.     

Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.     

Inertial methods for finding minimum-norm solutions of the split variational inequality problem beyond monotonicity

Numerical Algorithms - In solving the split variational inequality problems, very few methods have been considered in the literature and most of these few methods require the underlying operators...     

解变分不等式问题的混合方法

By using Fukushima‘s differentiable merit function,Taji,Fukushima and Ibaraki have given a globally convergent modified Newton method for the strongly monotone variational inequality problem and proved their method to be quadratically convergent under certain assumptions in 1993. In this paper a hybrid method for the variational inequality problem under the assumptions that the mapping F is continuously differentiable and its Jacobian matrix F(x) is positive definite for all x∈S rather than strongly monotone and that the set S is nonempty, polyhedral,closed and convex is proposed. Armijo-type line search and trust region strategies as well as Fukushima‘s differentiable merit function are incorporated into the method. It is then shown that the method is well defined and globally convergent and that,under the same assumptions as those of Taji et al. ,the method reduces to the basic Newton method and hence the rate of convergence is quadratic. Computational experiences show the efficiency of the proposed method.     

Modified subgradient extragradient method for variational inequality problems

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.     

A von Neumann alternating method for finding common solutions to variational inequalities

By modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space.     

A hybrid iteration scheme for equilibrium problems,variational inequality problems and common fixed point problems in Banach spaces

In this paper, we introduce an iterative process which converges strongly to a common element of a set of common fixed points of finite family of closed relatively quasi-nonexpansive mappings, the solution set of generalized equilibrium problem and the solution set of the variational inequality problem for an $\alpha$-inverse strongly monotone mapping in Banach spaces.     

Linesearch methods for bilevel split pseudomonotone variational inequality problems

Numerical Algorithms - In this paper, we propose Linesearch methods for solving a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level...     

Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems

The subgradient extragradient method can be considered as an improvement of the extragradient method for variational inequality problems for the class of monotone and Lipschitz continuous mappings. In this paper, we propose two new algorithms as combination between the subgradient extragradient method and Mann-like method for finding a common element of the solution set of a variational inequality and the fixed point set of a demicontractive mapping.     

Pseudomonotone variational inequality problems: Existence of solutions

Necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given. This research was partially done while the author was visiting the University of Chile thanks to the support of an ECOS program.     

General convergence analysis for two-step projection methods and applications to variational problems

First a general model for two-step projection methods is introduced and second it has been applied to the approximation solvability of a system of nonlinear variational inequality problems in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^0,y^0\in K$, compute sequences $\left\{x^k\right\}$ and $\left\{y^k\right\}$ such that $x^{k+1}=(1-a^k)x^k+a^k{P}_K[y^k-\rho T\left(y^k\right)]\text{for }\rho >0$$y^k=(1-b^k)x^k+b^k{P}_K[x^k-\eta T\left(x^k\right)]\text{for }\eta >0\text{,}$ where $T:K\to H$ is a nonlinear mapping on $K,P_K$ is the projection of $H$ onto $K$, and $0\le a^k,b^k\le 1$. The two-step model is applied to some variational inequality problems.     

New extragradient methods for solving variational inequality problems and fixed point problems

In this paper, we introduce two new iterative algorithms for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the variational inequality problem with a monotone and Lipschitz continuous mapping in real Hilbert spaces, by combining a modified Tseng’s extragradient scheme with the Mann approximation method. We prove weak and strong convergence theorems for the sequences generated by these iterative algorithms. The main advantages of our algorithms are that the construction of solution approximations and the proof of convergence of the algorithms are performed without the prior knowledge of the Lipschitz constant of cost operators. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.     

Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems

Whether or not the general asymmetric variational inequality problem can be formulated as a differentiable optimization problem has been an open question. This paper gives an affirmative answer to this question. We provide a new optimization problem formulation of the variational inequality problem, and show that its objective function is continuously differentiable whenever the mapping involved in the latter problem is continuously differentiable. We also show that under appropriate assumptions on the latter mapping, any stationary point of the optimization problem is a global optimal solution, and hence solves the variational inequality problem. We discuss descent methods for solving the equivalent optimization problem and comment on systems of nonlinear equations and nonlinear complementarity problems.     

Perturbed solutions of variational inequality problems over polyhedral sets

This paper is concerned with variational inequality problems defined over polyhedral sets, which provide a generalization of many diverse problems of mathematical programming, complementarity, and mathematical economics. Differentiability properties of locally unique perturbed solutions to such problems are studied. It is shown that, if a simple sufficient condition is satisfied, then the perturbed solution is locally unique, continuous, and directionally differentiable. Furthermore, under an additional regularity assumption, the perturbed solution is also continuously differentiable.     

Existence of solutions for generalized variational inequality problems in Banach spaces

In this paper, we prove the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over compact convex subsets in a reflexive Banach space with a Fréchet differentiable norm. Moreover, we give some conditions that guarantee the existence of solutions of generalized variational inequality for upper semicontinuous multivalued mappings with compact contractible values over unbounded closed convex subsets. The result obtained in this paper improves and extends the recent ones announced by Yu and Yang [J. Yu, H. Yang, Existence of solutions for generalized variational inequality problems, Nonlinear Anal., 71 (2009) e2327-e2330] and many others.     

Extragradient methods for differential variational inequality problems and linear complementarity systems

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed‐point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.