Abstract

One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well-known McCormick relaxation for a product of two variables $x$ and $y$ over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of $xy$ can be much tighter when computed over a strict, nonrectangular subset of the box. In order to exploit this in practice, we propose computing valid linear inequalities for the projection of the feasible region onto the $x$-$y$-space by solving a sequence of linear programs akin to optimization-based bound tightening. These valid inequalities allow us to employ results from the literature to strengthen the classical McCormick relaxation. As a consequence, we obtain a stronger convexification procedure that exploits problem structure and can benefit from supplementary information obtained during the branch-and-bound algorithm such as an objective cutoff. We complement this by a new bound tightening procedure that efficiently computes the best possible bounds for $x$, $y$, and $xy$ over the available projections. Our computational evaluations using the academic solver SCIP exhibit that the proposed methods are applicable to a large portion of the public test library MINLPLib and help to improve performance significantly.

Keywords

  1. mixed-integer quadratically constrained programs
  2. nonconvex
  3. global optimization
  4. separation
  5. propagation
  6. projection
  7. bilinear terms

MSC codes

  1. 90-08
  2. 90C27
  3. 90C26

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Information & Authors

Information

Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1339 - 1365
ISSN (online): 1095-7189

History

Submitted: 13 March 2019
Accepted: 10 January 2020
Published online: 11 May 2020

Keywords

  1. mixed-integer quadratically constrained programs
  2. nonconvex
  3. global optimization
  4. separation
  5. propagation
  6. projection
  7. bilinear terms

MSC codes

  1. 90-08
  2. 90C27
  3. 90C26

Authors

Affiliations

Funding Information

Bundesministerium für Wirtschaft und Energie : 03ET1549D
Bundesministerium für Bildung und Forschung https://doi.org/10.13039/501100002347 : 05M14ZAM

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