A lower bound for the complexity of linear optimization from a quantifier-elimination point of view

We discuss the impact of data structures in quantifier elimination. We analyze the arithmetic complexity of the feasibility problem in linear optimization theory as a quantifier-elimination problem. For the case of polyhedra defined by $2n$ halfspaces in $mathbb{R}^n$ we prove that, if dense representation is used to code polynomials, any quantifier-free formula expressing the set of parameters describing nonempty polyhedra has size $Omega(4^{n})$