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Complexity of optimizing over the integers

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Abstract

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of mixed-integer convex optimization, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities.

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Notes

  1. There is also a factor of \(\log (d)\) which shows up due to technical reasons of using \(\Vert \cdot \Vert _2\) instead of \(\Vert \cdot \Vert _\infty \).

  2. A subtlety here is to make sure that one has access to a separation oracle for the lower dimensional subproblems. This is not hard to implement given access to a separation oracle for C: given a point in the new space, one maps back to \(\mathbb R^n\times \mathbb R^d\) and queries the separation oracle there.

  3. There is a slight discrepancy because of the use of the \(\Vert \cdot \Vert _\infty \)-norm for the information complexity bound (see Theorem 4.2), and the use of \(\Vert \cdot \Vert _2\)-norm here. This adds a \(\log (d)\) factor to the complexity of the ellipsoid algorithm, compared to the information complexity bound. We are not aware of any work that resolves this discrepancy.

  4. There is a technical problem that arises here between the notions of algorithm and proof. We have omitted all discussions of cutting plane and branch-and-bound proofs here, which are powerful tools to prove unconditional lower bounds on these algorithms. The precise statement is that any branch-and-bound proof based on variable disjunctions can be replaced by a lift-and-project cutting plane proof of the same size. See [10] for details.

  5. “Lift-and-project cuts” here mean disjunctive cutting planes based on the variable disjunctions (typically the phrase “lift-and-project” is reserved for 0/1 problems).

  6. The same caveat as in point 1. regarding algorithms versus proofs applies.

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Acknowledgements

The author gratefully acknowledges support from Air Force Office of Scientific Research (AFOSR) grant FA95502010341 and National Science Foundation (NSF) grants CCF2006587.

The author benefited greatly from discussions with Daniel Dadush at CWI, Amsterdam and Timm Oertel at FAU, Erlangen-Nürmberg. Comments from two anonymous referees helped the author significantly to consolidate the material, improve its presentation and make tighter connections to the existing literature on the complexity of optimization.

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    Amitabh Basu

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Basu, A. Complexity of optimizing over the integers. Math. Program. 200, 739–780 (2023). https://doi.org/10.1007/s10107-022-01862-z

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