  In recent years, contour-based eigensolvers have emerged as a standard
approach for the solution of large and sparse eigenvalue problems. Building
upon recent performance improvements through non-linear least square
optimization of so-called rational filters, we introduce a systematic method to
design these filters by minimizing the worst-case convergence ratio and
eliminate the parametric dependence on weight functions. Further, we provide an
efficient way to deal with the box-constraints which play a central role for
the use of iterative linear solvers in contour-based eigensolvers. Indeed,
these parameter-free filters consistently minimize the number of iterations and
the number of FLOPs to reach convergence in the eigensolver. As a byproduct,
our rational filters allow for a simple solution to load balancing when the
solution of an interior eigenproblem is approached by the slicing of the sought
after spectral interval.
