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The complexity of approximating a nonlinear program

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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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Authors and Affiliations

  1. Advanced Networking Laboratory, IBM T.J. Watson Research Center, P.O. Box 704, 10598, Yorktown Heights, NY, USA

    Mihir Bellare

  2. Department of Computer Science, University of California at Davis, 95616, Davis, CA, USA

    Phillip Rogaway

Authors
  1. Mihir Bellare
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  2. Phillip Rogaway
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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

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