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Conic mixed-integer rounding cuts

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Abstract

A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints. Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either second-order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree. Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming. We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean–variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.

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Authors and Affiliations

  1. Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA, 94720-1777, USA

    Alper Atamtürk & Vishnu Narayanan

Authors
  1. Alper Atamtürk
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  2. Vishnu Narayanan
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Correspondence to Alper Atamtürk.

Additional information

This research has been supported, in part, by Grant # DMI0700203 from the National Science Foundation.

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Atamtürk, A., Narayanan, V. Conic mixed-integer rounding cuts. Math. Program. 122, 1–20 (2010). https://doi.org/10.1007/s10107-008-0239-4

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