An illumination problem with tradeoff between coverage of a dataset and aperture angle of a conic light beam

Let $\{a_i:i\in I\}$ be a finite set in ${‐{R}}^n$ . The illumination problem addressed in this work concerns the optimal location and orientation of a conic light beam $\vartheta =2arccoss$ of the conic light beam is a decreasing function of the sharpness coefficient $s\in [0,1]$ . The problem at hand is to select an apex z in a prescribed compact region $Z\subseteq {‐{R}}^n$ and a unit vector $y\in {‐{R}}^n$ so that the conic light beam R(z, y, s) fulfils two conflicting requirements: it captures as many points $a_i$ as possible and, at the same time, it has a sharpness coefficient s as large as possible.