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Finding point configurations, that yield the maximum polarization (Chebyshevconstant) is gaining interest in the field of geometric optimization. In thepresent article, we study the problem of unconstrained maximum polarization oncompact sets. In particular, we discuss necessary conditions for localoptimality, such as that a locally optimal configuration is always contained inthe convex hull of the respective darkest points. Building on this, we proposetwo sequences of mixed-integer linear programs in order to compute lower andupper bounds on the maximal polarization, where the lower bound isconstructive. Moreover, we prove the convergence of these sequences towards themaximal polarization.

Bounds on polarization problems on compact sets via mixed integer programming