A note on the combination of equilibrium problems

We show that the solution set of a strictly convex combination of equilibrium problems is not necessarily contained in the corresponding intersection of solution sets of equilibrium problems even if the bifunctions defining the equilibrium problems are continuous and monotone. As a consequence, we s...

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Bibliographic Details
Published in:Mathematical methods of operations research (Heidelberg, Germany) Germany), 2020-04, Vol.91 (2), p.311-323
Main Authors: Ha, Nguyen Thi Thanh, Thanh, Tran Thi Huyen, Hai, Nguyen Ngoc, Manh, Hy Duc, Van Dinh, Bui
Format: Article
Language:English
Subjects:
Business and Management
Calculus of Variations and Optimal Control
Optimization
Economic theory
Equilibrium
Mathematics
Mathematics and Statistics
Numerical methods
Operations research
Operations Research/Decision Theory
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Summary:We show that the solution set of a strictly convex combination of equilibrium problems is not necessarily contained in the corresponding intersection of solution sets of equilibrium problems even if the bifunctions defining the equilibrium problems are continuous and monotone. As a consequence, we show that some results given in some recent papers are not always true. Therefore different numerical methods for computing common solutions of families of equilibrium problems proposed in the literature may not converge under the monotonicity assumption. Finally, we prove that if the bifunctions are also parapseudomonotone, then the solution set of any strictly convex combination of a family of equilibrium problems is equivalent to the solution set of the intersection of the same family of equilibrium problems.
ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-019-00690-w