arXiv:2607.14519 2026 Optimization 1 ideas extracted · analyzed Aug 30, 2026
The paper provides a continuous-time constrained optimization mechanism in which the projection geometry is itself state dependent. Its transferable asset is that preconditioning is inserted inside the projected step, rather than applied after a fixed Euclidean projection, while the equilibrium remains a solution of the original variational inequality under positive-definite metrics. This suggests a constrained optimizer whose diagonal or low-rank metric adapts to the current parameter state and curvature, potentially improving convergence on anisotropic objectives without changing the target solution. The first implementation should use diagonal state-dependent metrics on box-constrained toy problems, where the weighted projection is exact and inexpensive.
Replace the usual projected gradient step with a relaxed projection in a positive-definite metric that changes with the current parameter state. The metric acts as a continuous preconditioner before projection, so updates can be large along poorly conditioned directions while remaining feasible.