Convergence theory for approximately separable objectives

Establish convergence guarantees for consensus-based optimization when the objective function is only approximately linearly separable, extending the guarantees available for exactly separable functions.

Background

The proposed method first estimates the coordinate directions defining a linearly separable objective and then applies consensus-based optimization in the estimated coordinate system. The recovery procedure provides approximation guarantees for the coordinate subspace, but the paper does not analyze how the resulting estimation error affects the convergence of the optimization stage.

The unresolved issue is whether convergence results analogous to those for anisotropic consensus-based optimization on exactly separable functions remain valid when the objective or its recovered representation is only approximately separable.