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 $x_n^\star \in \partial f^n(x_n)$ such that $\|x_n^\star\|_\star \to 0$. 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.