Tight subgradient-norm convergence conjecture for the proximal point algorithm

Let $f\in\mathcal{F}_{0,\infty}(\mathbb{R}^n)$, let $\{\alpha_i\}_{i\geq 1}$ be a sequence of positive step sizes, and let $B\succ 0$ define $\|x\|_B=\sqrt{x^{\mathsf T}Bx}$ and $\|g\|_{B^{-1}}=\sqrt{g^{\mathsf T}B^{-1}g}$. Let $x_*$ be an optimal point and let $x_0\in\mathbb{R}^n$ satisfy $\|x_0-x_*\|_B\leq R$. Let $\{x_i\}_{i\geq 1}$ be generated by the proximal point algorithm with these step sizes.

Subgradient-norm convergence conjecture. For every iterate $x_N$, there exists a subgradient $g_N\in\partial f(x_N)$ such that

$$\|g_N\|_{B^{-1}}\leq\frac{R}{\sum_{i=1}^{N}\alpha_i}.$$

In particular, the proximal-point subgradient $g_N=(x_{N-1}-x_N)/\alpha_N$ is a subgradient satisfying this inequality.

This conjecture concerns the tight convergence rate of the residual subgradient norm for the proximal point algorithm over the class of proper lower semicontinuous convex functions represented by $\mathcal{F}_{0,\infty}$. The preceding function-value rate is known to be tight, whereas the stated subgradient-norm bound is supported by numerical evidence and remains to be established in general.