  In biology, predicting RNA secondary structures plays a vital role in
determining its physical and chemical properties. Although we have powerful
energy models to predict them as well as parametric analysis to understand the
models themselves, the large number of parameters involved makes exploring the
parameter space and effective fine-tuning complicated at best. The literature
describes an approach via so-called RNA polytopes and several attempts to
compute them entirely, but computing explicitly the polytopes is both
practically and theoretically intractable. In this thesis, we demonstrate how
to further modify the dynamic programming algorithms used in RNA secondary
structure prediction, and more generally how to use only supporting functions
to gather some information about the polytopes without explicit construction.
We provide the mathematical frameworks with proofs or sketch thereof whenever
necessary, and carry out some numerical experiments to show that our proposed
methods are practical even when the number of parameters is large. As it turns
out, one of our methods provides a solution to another problem in computational
geometry previously unsolved to our knowledge, and we hope this thesis will
accommodate future studies in RNA, as well as inspire further researches on the
potential uses of polytopes' supporting functions in computational geometry.
