  We study how the lift-and-project procedure applies to the multiparametric
analysis of conic linear optimization (CLO) problems. We first introduce the
concept of a pair of primal and dual conic representable sets and define the
set-valued mappings between them. We then explore a novel kind of duality of
mpCLOs, which allows us to generalize as well as treat previous results for the
mulitparametric analysis in a unified framework. In particular, we discuss the
behavior of the optimal partition of a conic representable set. This leads to
the invariant region decomposition of a conic representable set that is more
general than the known results in the literatures. Finally, we study the
properties of the optimal objective values as a function of that parametric
vectors. All results are corroborated by examples having correlation.
