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On the complexity of finding a local minimizer of a quadratic function over a polytope

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Abstract

We show that unless P=NP, there cannot be a polynomial-time algorithm that finds a point within Euclidean distance \(c^n\) (for any constant \(c \ge 0\)) of a local minimizer of an n-variate quadratic function over a polytope. This result (even with \(c=0\)) answers a question of Pardalos and Vavasis that appeared in 1992 on a list of seven open problems in complexity theory for numerical optimization. Our proof technique also implies that the problem of deciding whether a quadratic function has a local minimizer over an (unbounded) polyhedron, and that of deciding if a quartic polynomial has a local minimizer are NP-hard.

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Notes

  1. This implies that these problems remain NP-hard even if the bitsize of all numerical data are \(O(\log n)\), where n is the number of variables. For a strongly NP-hard problem, even a pseudo-polynomial time algorithm—i.e., an algorithm whose running time is polynomial in the magnitude of the numerical data of the problem but not necessarily in their bitsize—cannot exist unless P = NP. See [4] or [1, Section 2] for more details.

  2. The converse of this statement also holds, but we do not need it for the proof of Theorem 2.1.

  3. This unique local (and therefore global) minimizer is guaranteed to have rational entries with polynomial bitsize; see [15].

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Authors and Affiliations

  1. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA

    Amir Ali Ahmadi

  2. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA

    Jeffrey Zhang

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  1. Amir Ali Ahmadi
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  2. Jeffrey Zhang
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Correspondence to Amir Ali Ahmadi.

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This work was partially supported by an AFOSR MURI award, the DARPA Young Faculty Award, the Princeton SEAS Innovation Award, the NSF CAREER Award, the Google Faculty Award, and the Sloan Fellowship.

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Ahmadi, A.A., Zhang, J. On the complexity of finding a local minimizer of a quadratic function over a polytope. Math. Program. 195, 783–792 (2022). https://doi.org/10.1007/s10107-021-01714-2

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  • DOI: https://doi.org/10.1007/s10107-021-01714-2

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