When Full-Space and Reduced-Space Methods Diverge in Equality-Constrained Optimization: A Conditioning Perspective

Equality-constrained optimization is central to modern physical AI, where desired feasible solutions are typically computed via Newton-type methods due to their rapid convergence. However, when selecting between the two primary formulations, reduced-space vs. full-space, practitioners routinely rely on domain-specific heuristics rather than a principled framework. In this work, we show that the practical boundary between these two approaches is fundamentally governed by the spectral profile of the state constraint Jacobian. While both approaches succeed in well-conditioned settings, their numerical behaviors diverge under ill-conditioning. Our theoretical analysis shows that reduced-space methods excel when ill-conditioning is driven by excessively large singular values, whereas full-space methods prove superior when driven by vanishing singular values. Empirical evaluations across four diverse benchmarks validate our theoretical findings.