Optimal Diagonal Preconditioning Beyond Worst-Case Conditioning: Theory and Practice of Omega Scaling

We study optimal diagonal preconditioning using the classical worst-case $\kappa$-condition number and the averaging-based $\omega$-condition number. For the $\kappa$-optimal preconditioning problem, we derive an affine-based pseudoconvex reformulation with three key advantages: all stationary points are global minima, subgradients are inexpensive to compute, and the optimization variable is an $n$-dimensional vector rather than an $n\times n$ matrix as in semidefinite programming (SDP) approaches. We develop a simple and highly efficient subgradient method, with convergence guarantees, for solving this pseudoconvex formulation that is substantially more scalable and accurate than existing SDP-based methods. For the $\omega$-condition number, we provide explicit characterizations of optimal diagonal and block diagonal preconditioners. In particular, we show that several classical preconditioners, including Jacobi and row/column normalization, are $\omega$-optimal, and that matrix balancing schemes monotonically reduce $\omega$ and converge to stationary points of the two-sided problem. To the best of our knowledge, this is the first unified and explicit characterization of optimality conditions for both $\kappa$ and $\omega$-based preconditioning. Our numerical experiments further reveal a striking phenomenon: although $\kappa$-optimal preconditioners achieve stronger reductions in the worst-case condition number, $\omega$-optimal preconditioners are substantially cheaper to compute and yield better performance for iterative methods such as preconditioned conjugate gradient (PCG) and least squares method (LSQR). Moreover, applying $\omega$-optimal scaling to linear systems that are already $\kappa$-optimally preconditioned leads to further improvements in PCG iterations.