Dimension reduction for semidefinite programs via Jordan algebras

TL;DR

This paper introduces a novel dimension reduction technique for semidefinite programs using Jordan algebra theory, enabling more efficient optimization by exploiting symmetry and algebraic structure.

Contribution

It develops a method to reduce SDP size via Jordan algebra-based symmetry reduction, including an algorithm for optimal rank minimization and block-diagonalization.

Findings

Effective dimension reduction demonstrated on examples

Algorithm achieves near-optimal block-diagonalization

Method extends to other symmetric cone programs

Abstract

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. Finally, we give combinatorial versions of our algorithm that execute at reduced computational cost and illustrate effectiveness of an implementation on examples. Through the theory of Jordan algebras, the proposed method easily extends to linear and second-order-cone programming and, more generally, symmetric cone optimization.