A Unifying View of Anchoring via Operator-Side Tikhonov Regularization

Anchored fixed point and monotone equation methods, including Halpern iteration, extra anchored gradient, and their relatives, add a vanishing pull toward a reference point to obtain last-iterate guarantees. Existing anchored variants often achieve sharp last-iterate guarantees, but from the update-level perspective the placement of the anchor can be algorithm-specific and conceptually opaque. We show that anchoring admits a single operator-side construction: regularize the operator queried by the base method with a vanishing Tikhonov term, then run the unmodified base method. Applied to the Picard iteration, this recipe reproduces the Halpern iteration; applied to the forward step, extragradient, and Popov methods, it yields three variants whose anchor placements inherit the base method's query pattern. The four analyses share a residual recurrence, recovering the \(O(1/k)\) Halpern residual-norm convergence rate, giving \(O(1/k)\) for the regularized forward step, and giving \(O(1/k)\) for the regularized extragradient and Popov variants in the unconstrained monotone Lipschitz setting.