Summary: We propose a unified framework of extragradient-type methods for solving pseudomonotone variational inequalities, which allows one to take different stepsize rules and requires the computation of only two projections at each iteration. It is shown that the modified extragradient method of A. N. Iusem
and B. F. Svaiter
[Optimization 42, No. 4, 309–321 (1977; Zbl 0891.90135
)] falls within this framework with a short stepsize and so does the method of M. V. Solodov
and B. F.Svaiter
[SIAM J. Control Optimization 37, No. 3, 765–776 (1999; Zbl 0959.49007
)] with a long stepsize. It is further demonstrated that the algorithmic framework is globally convergent under mild assumptions and is sublinearly convergent if in addition a projection-type error bound holds locally. Preliminary numerical experiments are reported.