Abstract
This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.
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Acknowledgements
This research has been partially supported by Grants MTM2008-06695-C03-02 and MTM2011-29064-C03-03 from MICINN/MINECO, Spain. The research of the second author is also partially supported by Fondecyt Project No 1110019 and ECOS-Conicyt project No C10E08.
The authors are indebted to Professor Marco A. López for his valuable hints and also for his comments about the backgrounds on the subject. The authors are also indebted to the anonymous referees and the associate editor for their constructive critical comments.
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Communicated by Réne Henrion.
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Cánovas, M.J., Hantoute, A., Parra, J. et al. Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization. J Optim Theory Appl 160, 111–126 (2014). https://doi.org/10.1007/s10957-013-0371-z
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DOI: https://doi.org/10.1007/s10957-013-0371-z