Joint optimization and spatial approximation error

Quantify jointly the optimization error and the spatial approximation error for the time-discrete finite-particle Consensus-Based Optimization algorithm as the Galerkin dimension tends to infinity, thereby connecting the discrete finite-dimensional implementation with the underlying infinite-dimensional optimization problem.

Background

The paper establishes convergence for a spatially truncated, time-discrete, finite-particle Consensus-Based Optimization system in a separable Hilbert space. Its theoretical guarantees control consensus formation and optimization quality over a fixed finite-dimensional active subspace, while the trace-class covariance provides bounds that are uniform in the active-space dimension.

The unresolved issue is to combine these two error sources—algorithmic optimization error and Galerkin spatial approximation error—in a single quantitative analysis as the active dimension increases without bound. Such a result would directly link the computable finite-dimensional algorithm to the original infinite-dimensional optimization problem.