Theorem 2 — EGRAAL converges to a solution of a monotone variational inequality

Let $\mathcal E$ be a finite-dimensional real inner product space, $g:\mathcal E\to(-\infty,+\infty]$ and $F:\operatorname{dom} g\to\mathcal E$, and assume:

1. (C1) the solution set $S$ of the variational inequality $\langle F(z^*),z-z^*\rangle+g(z)-g(z^*)\ge0$ for all $z\in\mathcal E$ is nonempty;
2. (C2) $g$ is proper, convex and lower semicontinuous;
3. (C3) $F$ is monotone on $\operatorname{dom} g$;
4. $F$ is locally Lipschitz continuous, in the form: Lipschitz on every bounded subset of $\operatorname{dom} g$.

Let $(z^k)$, $(\bar z^k)$ be generated by Algorithm 1 (the Explicit Golden Ratio Algorithm) with any $\phi\in(1,\varphi]$, $\bar\lambda>0$, $\lambda_0>0$ and any starting points $z^0,z^1$. Then there is $z^*\in S$ with

The theorem shows that a method with fully explicit, adaptive stepsizes — no linesearch and no knowledge of a Lipschitz constant — solves monotone variational inequalities with a merely locally Lipschitz operator.

Formalization Note The paper's "locally Lipschitz" is formalized as Lipschitz continuity on every bounded subset of $\operatorname{dom} g$, the property its proof (via Lemma 2) uses; the two agree when $\operatorname{dom} g$ is closed and the formal hypothesis is stronger otherwise. $F$ is a total map whose values off $\operatorname{dom} g$ are never used.

Retired 2026-10-27: a Lean-vs-paper fidelity audit found this statement weaker than, or narrower than, the cited result. A corrected statement will be resubmitted.