TWO-STAGE PROXIMAL ALGORITHMS FOR EQUILIBRIUM PROBLEMS IN HADAMARD SPACES
Abstract
We consider the equilibrium problems in the Hadamard
metric spaces. We obtained a theorem about weak convergence of
the two-stage proximal algorithm for pseudo-monotone equilibrium
programming problems in Hadamard spaces. We proposed an adaptive two-stage proximal algorithm for problems in metric Hadamard
spaces. The parameter update rule does not use the values of the
Lipschitz constants of the bifunction. In contrast to the rules of the
linear search type, it does not require calculations of the bifunction
values at additional points. For pseudo-monotone bifunctions of the
Lipschitz type, we prove the theorem on weak convergence of the
sequences generated by the algorithm. The adaptive extraproximal
algorithm is proposed and theoretically substantiated. A regularized adaptive extraproximal algorithm is proposed and theoretically
substantiated. We used the classical Halpern scheme to regularize the
basic extraproximal procedure. For pseudo-monotone bifunctions of
the Lipschitz type, we proved the convergence theorem for regularized adaptive extraproximal algorithm. We showed that the proposed
algorithm could be applied to pseudo-monotone ones of variational
inequalities in Hilbert spaces.
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