  In this paper we investigate the optimal partition approach for
multiparametric conic linear optimization (mpCLO) problems in which the
objective function depends linearly on vectors. We first establish more useful
properties of the set-valued mappings early given by us (arXiv:2006.08104) for
mpCLOs, including continuity, monotonicity and semialgebraic property. These
properties characterize the notions of so-called linearity and nonlinearity
subsets of a feasible set, which serve as stability regions of the partition of
a conic (linear inequality) representable set. We then use the arguments from
algebraic geometry to show that a semialgebraic conic representable set can be
decomposed into a union of finite linearity and/or nonlinearity subsets. As an
application, we investigate the boundary structure of the feasible set of
generic semialgebraic mpCLOs and obtain several nice structural results in this
direction, especially for the spectrahedon.
