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Unified framework of extragradient-type methods for pseudomonotone variational inequalities. (English) Zbl 1039.49014
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 {\it A. N. Iusem} and {\it 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 {\it M. V. Solodov} and {\it 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.

##### MSC:
 49J40 Variational methods including variational inequalities 90C33 Complementarity and equilibrium problems; variational inequalities (finite dimensions)
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