Mathematical Programming

, Volume 170, Issue 1, pp 141–176 | Cite as

Strong formulations for quadratic optimization with M-matrices and indicator variables

  • Alper AtamtürkEmail author
  • Andrés Gómez
Full Length Paper Series B


We study quadratic optimization with indicator variables and an M-matrix, i.e., a PSD matrix with non-positive off-diagonal entries, which arises directly in image segmentation and portfolio optimization with transaction costs, as well as a substructure of general quadratic optimization problems. We prove, under mild assumptions, that the minimization problem is solvable in polynomial time by showing its equivalence to a submodular minimization problem. To strengthen the formulation, we decompose the quadratic function into a sum of simple quadratic functions with at most two indicator variables each, and provide the convex-hull descriptions of these sets. We also describe strong conic quadratic valid inequalities. Preliminary computational experiments indicate that the proposed inequalities can substantially improve the strength of the continuous relaxations with respect to the standard perspective reformulation.


Quadratic optimization Submodularity Perspective formulation Conic quadratic cuts Convex piecewise nonlinear inequalities 

Mathematics Subject Classification

90C11 90C20 90C57 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Operations ResearchUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of Industrial Engineering , Swanson School of EngineeringUniversity of PittsburghPittsburghUSA

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