I am trying to implement a derivative of subgradient method that includes a distributed scenario and one of the requirements for convergence is to have a step size which satisifies the following conditions:
- $\sum_{k=1}^\infty\alpha_k = \infty$
- $\sum_{k=1}^\infty\alpha_k^2 < \infty$
- $\alpha_{k}\geq\alpha_{k+1}$
I have checked many references and an excellent candidate for step size rule is $\alpha_k=\frac{C}{k}$ where $C$ is a fixed constant. However, this rule appears to be not working as expected, since for a sufficiently small $C$, an optimal solution is not reached due to a fast decay in the updates concerning objective function. That is why I would like to know some alternatives to this rule that satisfies all 3 stated conditions.
I must clarify, this is not a simple subgradient method, I am implementing a distributed one, proposed in https://www.annualreviews.org/doi/full/10.1146/annurev-control-060117-105131, which has the update rule
$x_i(k + 1) = \Pi_X[y_i(k + 1) − \alpha_{k+1}g_i(k + 1)]$.
Where $\Pi_X$ is the projection operation to space $X$, $y_i$ is a weighted average of an agent's neighborhood and $g_i(k+1)$ is a subgradient at $y_i(k+1)$.