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We develop a practical semidefinite programming (SDP) facial reductionprocedure that utilizes computationally efficient approximations of thepositive semidefinite cone. The proposed method simplifies SDPs with nostrictly feasible solution (a frequent output of parsers) by solving a sequenceof easier optimization problems and could be a useful pre-processing techniquefor SDP solvers. We demonstrate effectiveness of the method on SDPs arising inpractice, and describe our publicly-available software implementation. We alsoshow how to find maximum rank matrices in our PSD cone approximations (whichhelps us find maximal simplifications), and we give a post-processing procedurefor dual solution recovery that generally applies to facial-reduction-basedpre-processing techniques. Finally, we show how approximations can be chosen topreserve problem sparsity.

Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone