Abstract
We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.
Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.
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Bellare, M., Rogaway, P. The complexity of approximating a nonlinear program. Mathematical Programming 69, 429–441 (1995). https://doi.org/10.1007/BF01585569
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DOI: https://doi.org/10.1007/BF01585569