  Resonances of Schr\"odinger Hamiltonians with point interactions are
considered. The main object under the study is the resonance free region under
the assumption that the centers, where the point interactions are located, are
known and the associated 'strength' parameters are unknown and allowed to bear
additional dissipative effects. To this end we consider the boundary of the
resonance free region as a Pareto optimal frontier and study the corresponding
optimization problem for resonances. It is shown that upper logarithmic bound
on resonances can be made uniform with respect to the strength parameters. The
necessary conditions on optimality are obtained in terms of first principal
minors of the characteristic determinant. We demonstrate the applicability of
these optimality conditions on the case of 4 equidistant centers by computing
explicitly the resonances of minimal decay for all frequencies. This example
shows that a resonance of minimal decay is not necessarily simple, and in some
cases it is generated by an infinite family of feasible resonators.
