Two logical hierarchies of optimization problems over the real numbers

We 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 [9],[8]). More precisely, based on a real analogue of Fagin's theorem [9] we deal with two classes MAX-NPℝ and MIN-NPℝ of maximization and minimization problems, respectively, and figure out their intrinsic logical structure. It is proven that MAX-NPℝ decomposes into four natural subclasses, whereas MIN-NPℝ decomposes into two. This gives a real number analogue of a result by Kolaitis and Thakur [10] in the Turing model. Our proofs mainly use techniques from [13]. Finally, approximation issues are briefly discussed.

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