Abstract: We present an overview of the main results on calmness in convex
semi-infinite optimization. The first part addresses the calmness of the
feasible set and the optimal set mappings for the linear semi-infinite
optimization problem in the setting of canonical perturbations, and also
in the framework of full perturbations. While there exists a clear
proportionality between the calmness moduli of the feasible set mappings
in both contexts, the analysis of the relationship between the calmness
moduli of the argmin mappings is much more complicated. Point-based
expressions (only involving the nominal problem’s data) for the calmness
moduli are provided. The second part focuses on convex semi-infinite
optimization, and provides a characterization of the Hölder calmness of
the optimal set mapping, by showing its equivalence with the Hölder
calmness of a certain (lower) level set mapping.
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