In this paper, we use a unified approach to analyze the local convergence behavior of a wide class of projection-type methods for solving variational inequality problems. Under certain conditions, it is shown that, in a finite number of iterations, either the sequence of iterates terminates at a solution of the concerned problem or all iterates enter and remain in the relative interior of the optimal face and, hence, the subproblem reduces to a simpler form.

In this paper, we study the equivalence characterizations of several modified fixedpoint equations to variational inequalities (VI). Based on these equations, we give some applications in constructing iterative methods for the solution of the VI. Especially, we show global convergence, the sublinear convergence, and the finite termination of a new iterative algorithm under certain conditions.

This paper develops convergence theory of the gradient projection method by Calamai andMore (Math. Programming, vol. 39, 93-116, 1987) which, for minimizing a continuously differentiable optimization problem min{f .(x): x ∈ Ω} where Ω is a nonempty closed convex set, generates a sequence xk+1=(ak-ak)▽f (xk))where the stepsize ak > 0 is chosen suitably. It is shown that, when f (x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either xk→x* and x* is a minimizer (stationary point); or ‖xk‖→arg min{f (x) : x ∈ Ω}= and f (xk) inf{ f (x): x ∈ Ω}.

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem.

For the extended linear complementarity problem over an affine subspace, we first study some characterizations of (strong) column/row monotonicity and (strong) R0-property. We then establish global s-type error bound for this problem with the column monotonicity or R0-property, especially for the one with the nondegeneracy and column monotonicity, and give several equivalent formulations of such error bound without the square root term for monotone affine variational inequality. Finally, we use this error bound to derive some properties of the iterative sequence produced by smoothing methods for solving such a problem under suitable assumptions.