Tightness of the Malitsky–Tam step-size bound

Let $A$ be maximally monotone and let $B$ be monotone and $L$ -Lipschitz on a Hilbert space. The forward-reflected-backward iteration is $x_{n+1}=J_{\lambda A}\bigl(x_n-2\lambda Bx_n+\lambda Bx_{n-1}\bigr)$ . Is the universal convergence restriction $0<\lambda<\frac{1}{2L}$ sharp, in the sense that convergence cannot be guaranteed for all such $(A,B)$ when $\lambda\geq\frac{1}{2L}$ ? The matched-skew instance $A=LJ$ , $B=LJ$ on $\mathbb{R}^2$ , where $J(x_1,x_2)=(-x_2,x_1)$ , fails to converge for every $\lambda\geq\frac{1}{2L}$ and diverges for every $\lambda>\frac{1}{2L}$ , thereby establishing sharpness.

References

Primary source

arXiv

Additional references

The sharp step-size constant for one-call reflection splittings on monotone inclusions — arXiv — Yekini Shehu

monotone-operator-theory optimization

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A September 2026 preprint claims to settle the sharpness question by exhibiting divergence at and above the recorded step-size threshold.

The problem asks whether the Malitsky–Tam step-size restriction can be improved. The latest report claims that a matched-skew example reaches the exact boundary and explains the extremal mechanism.

September 2026 sharp-threshold result

Yekini Shehu claims that the matched-skew instance fails to converge for every $\lambda \ge 1/(2L)$ and diverges for $\lambda > 1/(2L)$ ; a broader skew-rotation family is assigned an exact stability threshold. A separate technical note also claims tightness of the constrained bound and a larger sharp unconstrained threshold $1/(\sqrt{3}L)$ , but that formulation is not identified with the matched-skew result in the supplied material.

Current status (as of September 2026): The step-size bound is claimed to be sharp, including failure at $\lambda = 1/(2L)$ , but the available evidence is an unverified preprint claim.

Sources

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