  Algorithms for computing local minima of smooth objective functions enjoy a
mature theory as well as robust and efficient implementations. By comparison,
the theory and practice of saddle search is destitute. In this paper we present
results for idealized versions of the dimer and gentlest ascent (GAD) saddle
search algorithms that show-case the limitations of what is theoretically
achievable within the current class of saddle search algorithms: (1) we present
an improved estimate on the region of attraction of saddles; and (2) we
construct quasi-periodic solutions which indicate that it is impossible to
obtain globally convergent variants of dimer and GAD type algorithms.
