New Farkas-Type Results for Vector-Valued Functions: A Non-abstract Approach
This paper provides new Farkas-type results characterizing the inclusion of a given set, called contained set, into a second given set, called container set, both of them are subsets of some locally convex space, called decision space. The contained and the container sets are described here by means of vector functions from the decision space to other two locally convex spaces which are equipped with the partial ordering associated with given convex cones. These new Farkas lemmas are obtained via the complete characterization of the conic epigraphs of certain conjugate mappings which constitute the core of our approach. In contrast with a previous paper of three of the authors (Dinh et al. in J Optim Theory Appl 173:357–390, 2017), the aimed characterizations of the containment are expressed here in terms of the data.
KeywordsFarkas-type results Vector-valued functions Qualification conditions
Mathematics Subject Classification90C48 90C46 90C29
The authors wish to thank an anonymous referee for his/her valuable comments which helped to improve the manuscript. This research was supported by the National Foundation for Science & Technology Development (NAFOSTED) of Vietnam, Project 101.01-2015.27, Generalizations of Farkas lemma with applications to optimization, by the Ministry of Economy and Competitiveness of Spain and the European Regional Development Fund (ERDF) of the European Commission, Project MTM2014-59179-C2-1-P, and by the Australian Research Council, Project DP160100854.
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