SIAM Journal on Optimization, Volume 30, Issue 3, Page 1905-1921, January 2020.
The tight sublinear convergence rate of the proximal point algorithm for maximal monotone inclusion problems is established based on the squared fixed point residual. By using the performance estimation framework, the tight sublinear convergence rate problem is written as an infinite dimensional nonconvex optimization problem, which is then equivalently reformulated as a finite dimensional semidefinite programming (SDP) problem. By solving the SDP, the exact sublinear rate is computed numerically. Theoretically, by constructing a feasible solution to the dual SDP, an upper bound is obtained for the tight sublinear rate. On the other hand, an example in two dimensional space is constructed to provide a lower bound. The lower bound matches exactly the upper bound obtained from the dual SDP, which also coincides with the numerical rate computed. Hence, we have established the worst case sublinear convergence rate, which is tight in terms of both the order and the constants involved.

中文翻译：

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最大单调包含问题的近点算法的紧亚线性收敛速度

SIAM优化杂志，第30卷，第3期，第1905-1921页，2020年1月。
基于平方不动点残差，建立了最大单调包含问题的近点算法的紧亚线性收敛速度。通过使用性能估计框架，将紧亚线性收敛速率问题写为无限维非凸优化问题，然后将其等效地重新表示为有限维半定规划（SDP）问题。通过求解SDP，可以精确计算出次线性速率。从理论上讲，通过构造对偶SDP的可行解决方案，可以获得紧亚线性速率的上限。另一方面，二维空间中的示例被构造为提供下限。下限与从双SDP获得的上限精确匹配，这也与计算的数字速率一致。