References

Given a nonempty subset V of a metric space X and a function I : X → (−∞,∞] which is a proper extended real-valued function, consider well-posedness of the following abstract minimization problem:

min I(v), v ∈ V,

which we denote by (V, I). Let vV (I) := inf{I(v) : v ∈ V } denote the optimal value function. We assume I to be lower bounded on V, i.e., vV (I) > −∞, and let argminV (I) denote the (possibly void) set {v ∈ V : I(v) = vV (I)} of optimal solutions of problem (V, I). For ≥ 0, let us also denote by - argminV (I) the nonempty set {v ∈ V : I(v) ≤ vV (I) + } of -approximate minimizers of I. Recall (cf., e.g., [4], p.1) that problem (V, I) is said to be Tykhonov wellposed if I has a unique global minimizer on V towards which every minimizing sequence (i.e., a sequence {vn} ⊂ V, such that I(vn)→ vV (I)) converges. Put differently, there exists a point v0 ∈ V such that argminV (I) = {v0}, and whenever a sequence {vn} ⊂ V is such that Ivn → Iv0, one has vn → v0. The concept of Tykhonov wellposedness has been extended to minimization problems admitting non-unique optimal solutions. For our purpose in this paper, the most appropriate well-posedness notion for such problems is the one introduced in Bednarczuk and Penot [2] (cf. also [4], p.26):

Problem (V, I) is called metrically well-set (or M-well set) if argminV (I) = ∅ and for every minimizing sequence {vn}, one has

dist(vn, argmin V

(I))→ 0 as n →∞.