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Complexity, exactness, and rationality in polynomial optimization

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Abstract

We focus on rational solutions or nearly-feasible rational solutions that serve as certificates of feasibility for polynomial optimization problems. We show that, under some separability conditions, certain cubic polynomially constrained sets admit rational solutions. However, we show in other cases that it is NP Hard to detect if rational solutions exist or if they exist of any reasonable size. We extend this idea to various settings including near feasible, but super optimal solutions and detecting rational rays on which a cubic function is unbounded. Lastly, we show that in fixed dimension, the feasibility problem over a set defined by polynomial inequalities is in NP by providing a simple certificate to verify feasibility. We conclude with several related examples of irrationality and encoding size issues in QCQPs and SOCPs.

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Notes

  1. Throughout we will use the concept of size, or bit encoding length, of rational numbers, vectors, linear inequalities, and formulations. For these standard definitions we refer the reader to Section 2.1 in [32].

  2. Constraint (5g) as written is cubic, but is equivalent to three quadratic constraints by defining new variables \(y_{12} = y_1^2, \quad y_{22} = y_2^2\), and rewriting the constraint as \(-n^5 \sum _{j = 1}^{n} x_j^2 \, + \, y_{12} y_1 + y_{22} y_2 - 6 y_1y_2 + 4 \, - \, s \ \le \ -n^6\).

  3. We stress that (7) is a maximization problem.

  4. A numerical solver might yield a vector \(y^*\) that is near-binary, i.e. within a small tolerance. The analysis below is easily adjusted to handle such an eventuality.

  5. Recall that \(y^*\) is binary.

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Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Columbia University, New York, USA

    Daniel Bienstock

  2. Department of Industrial and Systems Engineering, Wisconsin Institute for Discovery, University of Wisconsin-Madison, Madison, USA

    Alberto Del Pia

  3. Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, USA

    Robert Hildebrand

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  2. Alberto Del Pia
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A. Del Pia is partially funded by ONR Grant N00014-19-1-2322. D. Bienstock is partially funded by ONR Grant N00014-16-1-2889. R. Hildebrand is partially funded by ONR Grant N00014-20-1-2156 and by AFOSR Grant FA9550-21-0107. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research or the Air Force Office of Scientific Research.

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Bienstock, D., Pia, A.D. & Hildebrand, R. Complexity, exactness, and rationality in polynomial optimization. Math. Program. 197, 661–692 (2023). https://doi.org/10.1007/s10107-022-01818-3

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