A modification of projection methods in finite dimensional spaces for symmetric variational inequalities of type

${(x-{x}^{*})}^{T}(H{x}^{*}+C)\ge 0,\phantom{\rule{0.166667em}{0ex}}\forall x\in {\Omega}$ with nonempty closed convex set

${\Omega}$ is considered. A known iterative method which bases on an equivalent fixed point formulation

$x={P}_{{\Omega}}(x-\beta (Hx+c))$ is modified by replacing the constant

$\beta >0$ by parameters

${\beta}_{k}$ which are adapted to the iterates

${x}^{k}$. A convergence theorem is established and numerical examples are given. However, in the experiments the earlier restrictions for the choice of

${\beta}_{k}$ are relaxed.