  In this paper, we introduce two golden ratio algorithms with new stepsize
rules for solving pseudomonotone and Lipschitz variational inequalities in
finite dimensional Hilbert spaces. The presented stepsize rules allow the
resulting algorithms to work without the prior knowledge of the Lipschitz
constant of operator. The first algorithm uses a sequence of stepsizes which is
previously chosen, diminishing and non-summable. While the stepsizes in the
second one are updated at each iteration and by a simple computation. A special
point is that the sequence of stepsizes generated by the second algorithm is
separated from zero. The convergence as well as the convergence rate of the
proposed algorithms are established under some standard conditions. Also, we
give several numerical results to show the behavior of the algorithms in
comparisons with other algorithms.
