Uniqueness of OptISTA in composite convex minimization with a proximable term

Determine whether OptISTA is the uniquely minimax optimal deterministic first-order algorithm for composite convex minimization with a proximable term, i.e., ascertain if any other deterministic first-order method achieves the exact optimal worst-case rate on this problem class.

Background

The paper references OptISTA as achieving exact optimality for composite convex minimization with a proximable term. However, the authors explicitly state that it is unknown if OptISTA is uniquely optimal; they further note reported numerical evidence suggesting non-uniqueness.

This suggests a broader open question of characterization of all exact optimal algorithms in the composite setting, analogous to the complete characterization achieved in the paper for nonexpansive fixed-point problems.