For quadratic cone programs with strongly convex objectives, choosing the objective scaling factor as sqrt(sigma_min / 2) provably minimizes the KKT condition number, and packaging this with hypersphere and row-normalization steps yields a fast, factorization-free preconditioner.
Projecting onto the intersection of a cone and a sphere,
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Optimal Preconditioning for Online Quadratic Cone Programming
For quadratic cone programs with strongly convex objectives, choosing the objective scaling factor as sqrt(sigma_min / 2) provably minimizes the KKT condition number, and packaging this with hypersphere and row-normalization steps yields a fast, factorization-free preconditioner.