Bounded Diagonally Stationary Sequences in Convex Optimization
Bernard Lemaire · Journal of convex analysis · 1994
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) x n ⋆ ∈ ∂ f n ( x n ) 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.