Two logical hierarchies of optimization problems over the real numbers ## AbstractWe introduce and study certain classes of optimization problems over the real numbers. The classes are defined by logical means, relying on metafinite model theory for so called $\R$-structures (see \cite{Graedel-Meer}, \cite{Graedel-Gurevich}). More precisely, based on a real analogue of Fagin's theorem \cite{Graedel-Meer} we deal with two classes $\maxnr$ and $\minnr$ of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that $\maxnr$ decomposes into four natural subclasses, whereas $\minnr$ decomposes into two. This gives a real number analogue of a result by Kolaitis and Thakur \cite{Kolaitis} in the Turing model. Our proofs mainly use techniques from \cite{Meer}. Finally, approximation issues are briefly discussed.
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