A projected subgradient method for solving generalized mixed variational inequalities

We consider the projected subgradient method for solving generalized mixed variational inequalities. In each step, we choose an ε_k-subgradient u^k of the function f and w^k in a set-valued mapping T, followed by an orthogonal projection onto the feasible set. We prove that the sequence is weakly convergent.     

A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces

The purpose of this paper is by using the generalized projection approach to introduce an iterative scheme for finding a solution to a system of generalized nonlinear variational inequality problem. Under suitable conditions, some existence and strong convergence theorems are established in uniformly smooth and strictly convex Banach spaces. The results presented in the paper improve and extend some recent results.     

A modified projection method for solving co-coercive variational inequalities

This paper presents a modified projection method for solving variational inequalities, which can be viewed as an improvement of the method of Yan, Han and Sun [X.H. Yan, D.R. Han, W.Y. Sun, A modified projection method with a new direction for solving variational inequalities, Applied Mathematics and Computation 211 (2009) 118-129], by adopting a new prediction step. Under the same assumptions, we establish the global convergence of the proposed algorithm. Some preliminary computational results are reported.     

A self-adaptive method for solving general mixed variational inequalities

The general mixed variational inequality containing a nonlinear term φ is a useful and an important generalization of variational inequalities. The projection method cannot be applied to solve this problem due to the presence of the nonlinear term. To overcome this disadvantage, Noor [M.A. Noor, Pseudomonotone general mixed variational inequalities, Appl. Math. Comput. 141 (2003) 529-540] used the resolvent equations technique to suggest and analyze an iterative method for solving general mixed variational inequalities. In this paper, we present a new self-adaptive iterative method which can be viewed as a refinement and improvement of the method of Noor. Global convergence of the new method is proved under the same assumptions as Noor's method. Some preliminary computational results are given.     

Homotopy method for solving ball-constrained variational inequalities

In this paper, we present a smoothing homotopy method for solving ball-constrained variational inequalities by utilizing a similar Chen-Harker-Kanzow-Smale function to smooth Robinson’s normal equation. Without any monotonicity condition on the defining map F, for the starting point chosen almost everywhere in Rⁿ, the existence and convergence of the homotopy pathway are proven. Numerical experiments illustrate that the method is feasible and effective.     

Modified descent-projection method for solving variational inequalities

In this paper, we propose a modified descent-projection method for solving variational inequalities. The method makes use of a descent direction to produce the new iterate and can be viewed as an improvement of the descent-projection method by using a new step size. Under certain conditions, the global convergence of the proposed method is proved. In order to demonstrate the efficiency of the proposed method, we provide numerical results for a traffic equilibrium problems.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

The penalty method for generalized multivalued nonlinear variational inequalities in Banach spaces

We consider the penalty method for solving generalized nonlinear variational inequalities. We obtain some existence theorems for the variational inequalities by the penalty method in reflexive real Banach spaces.     

Strongly nonlinear variational inequalities and generalized pseudocontractions

The solvability of a class of generalized strongly nonlinear variational inequality problems on nonempty closed convex sets in Hilbert spaces is presented.     

A posteriori error estimation for nonlinear variational problems by duality theory

In this paper, we use the duality theory of the calculus of variations to derive a posteriori error estimates. We obtain a general form of this (duality) error estimate and show that the known classes of a posteriori error estimates are its particular cases. Bibliography: 21 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 201–214. Translated by S. I. Repin.     

Projection iterative method for solving general variational inequalities

In this paper, we suggest and analyze a new projection iterative method for solving general variational inequalities by using a new step size. We also prove the global convergence of the proposed method under some suitable conditions. Some preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

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.     

An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems

In this paper, we introduce an inertial subgradient-type algorithm to find the common element of fixed point set of a family of nonexpansive mappings and the solution set of the single-valued variational inequality problem. Under the assumption that the mapping is monotone and Lipschitz continuous, we show that the sequence generated by our algorithm converges strongly to some common element of the fixed set and the solution set. Moreover, preliminary numerical experiments are also reported.     

A descent algorithm for solving monotone variational inequalities and monotone complementarity problems

The paper provides a descent algorithm for solving certain monotone variational inequalities and shows how this algorithm may be used for solving certain monotone complementarity problems. Convergence is proved under natural monotonicity and smoothness conditions; neither symmetry nor strict monotonicity is required.The author is grateful to two anonymous referees for their very valuable comments on an earlier draft of this paper.