Abstract
Using the recently proposed maximal quadratic-free sets and the well-known monoidal strengthening procedure, we show how to improve intersection cuts for quadratically-constrained optimization problems by exploiting integrality requirements. We provide an explicit construction that allows an efficient implementation of the strengthened cuts along with computational results showing their improvements over the standard intersection cuts. We also show that, in our setting, there is unique lifting which implies that our strengthening procedure is generating the best possible cut coefficients for the integer variables.
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Notes
- 1.
With a slight abuse of notation, we refer to a non-convex set \(C - M\) as S-free whenever the convex set \(C - m\) is S-free for every \(m \in M\).
- 2.
A function \(\psi \) is subadditive if \(\psi (x+y) \le \psi (x) + \psi (y)\).
- 3.
This means that if \(\rho \) is such that \(C = \{ s \in \mathbb {R}^p \,:\,\rho (s - f) \le 1\}\) then \(\rho (s) \ge \phi (s)\).
- 4.
This citation deals with a particular set S, but the proof can be easily extended to any conic set S.
- 5.
Note that \(S^{g}\) is contained on a halfspace, so \(S^{g}\)-freeness is with respect to the induced topology in \(H\).
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Chmiela, A., Muñoz, G., Serrano, F. (2023). Monoidal Strengthening and Unique Lifting in MIQCPs. In: Del Pia, A., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2023. Lecture Notes in Computer Science, vol 13904. Springer, Cham. https://doi.org/10.1007/978-3-031-32726-1_7
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