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Splitting methods for pseudomonotone mixed variational inequalities. (English) Zbl 0966.49011
The author analyzes some algorithms for solving a monotone mixed variational inequality Tu,v-u+ϕ(v)-ϕ(u)0, for allvH with a Hilbert space H, a nonlinear operator T:HH and a lower semicontinuous function ϕ:HR{+}· The resolvent operator technique is used to study a number of splitting methods for solving the mixed variational inequalities. The methods are based on the fact that a solution uH of a variational inequality satisfies the relation u=J ϕ [u-ρTu], where J ϕ =(I+ρϕ) -1 is the resolvent operator. The resolvent operator method requires the assumption that T must be strongly monotone for the convergence. In order to overcome this difficulty the author suggests the double resolvent formula u=J ϕ u - ρ T J ϕ [u-ρTu]· Several splitting methods for solving the iteration method connected with fixed point algorithms are analyzed.
MSC:
49J40Variational methods including variational inequalities