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We propose a new method for simplifying semidefinite programs (SDP) inspiredby symmetry reduction. Specifically, we show if an orthogonal projection mapsatisfies certain invariance conditions, restricting to its range yields anequivalent primal-dual pair over a lower-dimensional symmetric cone---namely,the cone-of-squares of a Jordan subalgebra of symmetric matrices. We present asimple algorithm for minimizing the rank of this projection and hence thedimension of this subalgebra. We also show that minimizing rank optimizes thedirect-sum decomposition of the algebra into simple ideals, yielding an optimal"block-diagonalization" of the SDP. Finally, we give combinatorial versions ofour algorithm that execute at reduced computational cost and illustrateeffectiveness of an implementation on examples. Through the theory of Jordanalgebras, the proposed method easily extends to linear and second-order-coneprogramming and, more generally, symmetric cone optimization.

Dimension reduction for semidefinite programs via Jordan algebras