Iterative algorithms for variational inequalities governed by boundedly lipschitzian and strongly monotone operators

Consider the variational inequality vi(c,f) of finding a point x∗ ∈ c satisfying the property 〈fx∗,x-x∗〉≥0 for all x ∈ c,where c is a level set of a convex function defined on a real hilbert space h and f:h→h is a boundedly lipschitzian (i.e.,lipschitzian on bounded subsets of h) and strongly monotone operator. he and xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see he and xu,2009). in this paper,relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. since our algorithms avoid calculating the projection pc (calculating pc by computing a sequence of projections onto half-spaces containing the original domain c) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded lipschitz constants of f (i.e.,lipschitz constants on some bounded subsets of h)),the implementations of our algorithms are very easy. the algorithms in this paper improve and extend the corresponding results of he and xu. © 2015 caiping yang and songnian he.