# Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints

@article{Bertsimas2020MixedProjectionCO, title={Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints}, author={Dimitris Bertsimas and Ryan Cory-Wright and Jean Pauphilet}, journal={ArXiv}, year={2020}, volume={abs/2009.10395} }

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model rank constraints. By leveraging regularization and strong duality, we prove that this modeling paradigm yields tractable convex optimization problems over the non-convex set of orthogonal projection matrices. Furthermore, we design outer-approximation…

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