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Abstract
The paper proposes a new shrinking gradient-like projection method for solving equilibrium problems. The algorithm combines the generalized gradient-like projection method with the monotone hybrid method. Only one optimization program is solved onto the feasible set at each iteration in our algorithm without any extra-step dealing with the feasible set. The absence of an optimization problem in the algorithm is explained by constructing slightly different cutting-halfspace in the monotone hybrid method. Theorem of strong convergence is established under standard assumptions imposed on equilibrium bifunctions. An application of the proposed algorithm to multivalued variational inequality problems (MVIP) is presented. Finally, another algorithm is introduced for MVIPs in which we only use a value of main operator at the current approximation to construct the next approximation. Some preliminary numerical experiments are implemented to illustrate the convergence and computational performance of our algorithms over others.
Acknowledgements
The author would like to thank the associate editor and two referees for their valuable comments and suggestions which helped us very much in improving and presenting the original version of this paper. The guidance of Profs. P.K. Anh and L.D. Muu is gratefully acknowledged.
Notes
No potential conflict of interest was reported by the author.
1 Two matrices P, Q are randomly generated as follows: we randomly choose . Set
,
as two diagonal matrices with eigenvalues
and
, respectively. Then, we make a positive semidefinite matrix Q and a negative semidefinite matrix T by using
and
with two random orthogonal matrices, respectively. Finally, set
.