Dimension reduction for semidefinite programs via Jordan algebras

Understand

We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction.

• Specifically, we show if an orthogonal projection map satisfies certain invariance conditions, restricting to its range yields an equivalent primal-dual pair over a lower-dimensional symmetric cone---namely, the cone-of-squares of a Jordan subalgebra of symmetric matrices.
• We present a simple algorithm for minimizing the rank of this projection and hence the dimension of this subalgebra.
• We also show that minimizing rank optimizes the direct-sum decomposition of the algebra into simple ideals, yielding an optimal "block-diagonalization" of the SDP.