Abstract
In this paper, we propose an approach to characterizing \({\epsilon} \)-solution sets of convex programs with a given \({\epsilon} >0\). The results are divided into two parts. The first one is devoted to establishing the expressions of \({\epsilon} \)-solution sets of a class of convex infinite programs. The representation is given based on the study of relationships among the following three sets: the set of Lagrange multipliers corresponding to a given \({\epsilon} \)-solution, the set of \({\epsilon} \)-solutions of the dual problem corresponding, and the set of \({\epsilon} \)-Kuhn–Tucker vectors associated with the problem in consideration. The second one is devoted to some special cases: the \({\epsilon} \)-solution sets of convex programs that have set constraints and the almost \({\epsilon} \)-solution sets of convex programs that have finite convex constraints. Several examples are given.
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Acknowledgements
The authors would like to thank the two anonymous reviewers, whose suggestions and comments improved the paper. The research of the first author was funded by Vietnam Ministry of Education and Training under Grant number B2021-SP2-06. A part of this work was done while he was visiting Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank VIASM for support and hospitality. The research of the second author was supported partially by the Grant MOST 108-2115-M-037-001, and the grant from Research Center for Nonlinear Analysis and Optimization Kaohsiung Medical University, Taiwan. The research of the third author was supported partially by Saigon University. He would also like to express his sincere thanks to the Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, where parts of his research were carried out.
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Tuyen, N.V., Wen, CF. & Son, T.Q. An approach to characterizing \(\epsilon \)-solution sets of convex programs. TOP 30, 249–269 (2022). https://doi.org/10.1007/s11750-021-00616-y
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DOI: https://doi.org/10.1007/s11750-021-00616-y
Keywords
- \({\epsilon} \)-solution
- \({\epsilon} \)-solution set
- Minimizing sequence
- \({\epsilon} \)-Kuhn–Tucker vector