Bounded Diagonally Stationary Sequences in Convex Optimization

Let X be a real normed linear space, f, fⁿ, n ∈ ℕ, be extended real-valued proper closed convex functions on X. A sequence xₙ in X is called diagonally stationary for fⁿ if for all n there exists

such that

\|x_n^\star\|_\star \to 0 ∥ x n ⋆ ∥ ⋆ → 0

. Such sequences arise in approximation methods for the problem of minimizing f. We present some general quantitative convergence results based upon metric variational convergence theory, appropriate equi-well-posedness and conditioning concepts for the limit function f, and Fejér monotonicity.