Abstract
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer assignments is finite. We develop a characterization for the more general case of unbounded integer variables together with a simple necessary condition for representability which we use to prove the first known negative results. Finally, we study representability of subsets of the natural numbers, developing insight towards a more complete understanding of what modeling power can be gained by using convex sets instead of polyhedral sets; the latter case has been completely characterized in the context of mixed-integer linear optimization.
M. Lubin and I. Zadik—Contributed equally to this work.
J.P. Vielma—Supported by NSF under grant CMMI-1351619.
We acknowledge the anonymous referees for improving the presentation of this work.
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Lubin, M., Zadik, I., Vielma, J.P. (2017). Mixed-Integer Convex Representability. In: Eisenbrand, F., Koenemann, J. (eds) Integer Programming and Combinatorial Optimization. IPCO 2017. Lecture Notes in Computer Science(), vol 10328. Springer, Cham. https://doi.org/10.1007/978-3-319-59250-3_32
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DOI: https://doi.org/10.1007/978-3-319-59250-3_32
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