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A graph-based decomposition method for convex quadratic optimization with indicators

Abstract

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This enables us to construct a compact extended formulation for the closure of the convex hull of the epigraph of the mixed-integer convex problem. Furthermore, motivated by inference problems with graphical models, we propose a novel decomposition method for a class of general (sparse) strictly diagonally dominant Q, which leverages the efficient algorithm for the path case. Our computational experiments demonstrate the effectiveness of the proposed method compared to state-of-the-art mixed-integer optimization solvers.

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Notes

  1. They consider a slightly different term, where the sparsity is imposed via a cardinality constraint \(a^\top z\le k\) instead of a penalization in the objective.

  2. For step size \(s_k=1/k\), we modify line 6 of Algorithm 2 to \((\alpha ,\beta )\leftarrow (\alpha ,\beta )+s_k \rho (\bar{x},\bar{z})\) (without normalization), since this version performed better in our computations.

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Acknowledgements

We thank the AE and the referees whose comments improved this paper.

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Correspondence to Simge Küçükyavuz.

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This research is supported, in part, by NSF grants 2006762, 2007814, 2152776, and ONR grant N00014-22-1-2127.

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Liu, P., Fattahi, S., Gómez, A. et al. A graph-based decomposition method for convex quadratic optimization with indicators. Math. Program. (2022). https://doi.org/10.1007/s10107-022-01845-0

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  • DOI: https://doi.org/10.1007/s10107-022-01845-0

Keywords

  • Quadratic optimization
  • Indicator variables
  • Sparsity
  • Decomposition
  • Graphical models
  • Fenchel dual
  • Convex hull

Mathematics Subject Classification

  • 90C11 (Mixed-integer optimization)
  • 49M27 (decomposition methods)
  • 90C25 (convex optimization)