Exact Parameter Regions for Generalized Optimistic Gradient Methods

The generalized optimistic gradient method (GOGD) is a one-call method for operator equations that includes the optimistic gradient method as the special case of equal step and correction parameters, yet its full admissible parameter region has remained unknown. For $L$-Lipschitz operators satisfying a weak-Minty condition with parameter $\rho$ and $-1\le \rho L<1$, we characterize exactly the parameter pairs that guarantee bounded iterates with square-summable increments and residuals, and show by a two-dimensional linear counterexample that no other pair does. In particular, equal step and correction parameters are admissible only for small $\rho L$. For monotone operators, the region is exactly the set of parameters yielding weak convergence to a zero from every initialization; for weak-Minty operators, weak convergence follows under standard demiclosedness and solution-set assumptions, with norm convergence in finite dimensions. The proof combines a Lyapunov identity with a matching spectral obstruction and yields best-iterate $O(1/N)$ bounds for squared residuals and increments. Under the stronger pairwise condition of co-hypomonotonicity, we further obtain last-iterate $o(1/N)$ rates on a subregion, and on the full region for affine operators. The new region substantially enlarges previously known sufficient regions, especially as $\rho L\to 1$: on a quadratic minimax example with $\rho L=0.9$, a parameter rule from the new region uses about $17$ times fewer operator evaluations than tuned candidates from earlier regions.