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On the complexity of detecting convexity over a box

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Abstract

It has recently been shown that the problem of testing global convexity of polynomials of degree four is strongly NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity only over a compact region, most commonly a box (i.e., a hyper-rectangle). In this paper, we show that this problem is also strongly NP-hard, in fact for polynomials of degree as low as three. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control.

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Notes

  1. Note that the instances of not all of NP-hard problems involve numerical data; consider, e.g., SATISFIABILITY [12]. For such problems, NP-hardness is by default in the strong sense.

  2. Here, the notation \(M\succeq 0\) is used to denote that a symmetric matrix M is positive semidefinite, i.e., has nonnegative eigenvalues. We will also use the notation \(A \succeq B\) for two symmetric matrices A and B to denote that \(A-B \succeq 0\).

  3. Recall that a graph is simple if it is unweighted, undirected, and has no self-loops or multiple edges.

  4. Note that the proof of Theorem 2.3 established strong NP-hardness of this problem as the box that arose in the proof was the unit hypercube, which is full-dimensional.

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Acknowledgements

We are grateful to two anonymous referees whose detailed and constructive feedback has improved this paper significantly.

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Authors and Affiliations

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

    Amir Ali Ahmadi

  2. Decision Sciences, INSEAD, Boulevard de Constance, Fontainebleau, France

    Georgina Hall

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  1. Amir Ali Ahmadi
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  2. Georgina Hall
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Correspondence to Georgina Hall.

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This work was partially supported by the DARPA Young Faculty Award, the CAREER Award of the NSF, the Google Faculty Award, the Innovation Award of the School of Engineering and Applied Sciences at Princeton University, and the Sloan Fellowship.

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Ahmadi, A.A., Hall, G. On the complexity of detecting convexity over a box. Math. Program. 182, 429–443 (2020). https://doi.org/10.1007/s10107-019-01396-x

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