A Quasiconvex Asymptotic Function with Applications in Optimization
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We introduce a new asymptotic function, which is mainly adapted to quasiconvex functions. We establish several properties and calculus rules for this concept and compare it to previous notions of generalized asymptotic functions. Finally, we apply our new definition to quasiconvex optimization problems: we characterize the boundedness of the function, and the nonemptiness and compactness of the set of minimizers. We also provide a sufficient condition for the closedness of the image of a nonempty closed and convex set via a vector-valued function.
KeywordsAsymptotic cones Asymptotic functions Quasiconvexity Nonconvex optimization Closedness criteria
Mathematics Subject Classification90C25 90C26 90C30
Part of this research was carried out during a stay of the first author in the Departamento de Ingeniería Matemática of the Universidad de Concepción, Chile, supported in part by Cooperación Internacional of Conicyt–Chile through Fondecyt 115-0973. The author wishes to thank the Department for its hospitality. For the second author, this research was partially supported by Conicyt–Chile under project Fondecyt Postdoctorado 3160205. Part of this work was carried out when the second author was visiting the Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, Barcelona, Spain, during November of 2016. The author wishes to thank the Department for its hospitality. The third author was supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and by the Severo Ochoa Programme for Centres of Excellence in R&D [SEV-2015-0563]. He is affiliated with MOVE (Markets, Organizations and Votes in Economics).
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