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On the stability of closed-convex-valued mappings and the associated boundaries

“…From (3.1) we get dom F ⊂ int Y and so the stability properties of F andF coincide for every y 0 ∈ dom F , and the same is true for their respective boundary mappings. In particular, from the results on the stability ofF andB at y 0 ∈ dom F in [2] and [3], we can establish the following facts on the stability of F and B at y 0 ∈ dom F = dom B:…”

“…There exists a wide literature on the stability ofF : (R n+1 ) T ⇒ R n when (R n+1 ) T is equipped with the pseudometric of the uniform convergence (see [3] and references therein). By Lemma 2 in [2], given y ∈ domF , F(y) = conv bdF(y) if and only if there exist indexes s, u ∈ T and a point x ∈ R n such that…”

“…The set Y := y ∈ (R n+1 ) T |F(y) = conv bdF(y) contains almost all the elements of (R n+1 ) T in a topological sense (Proposition 2 in [2]). Obviously, the restriction ofF to Y , say F , satisfies (1.1).…”

“…i.e., F(y) is a closed convex set which is neither the whole space R n nor a halfspace [2,Lemma 2]. This condition is usually satisfied by the feasible and the optimal set mappings of (ordinary and semi-infinite) parametric convex optimization problems.…”

Characterization of total ill-posedness in linear semi-infinite optimization

“…In [1] three notions of stability in linear programming are studied, and in [6] their equivalence in the LSIP setting is proven. A selection of different contributions to the stability of the general LSIP problem, when all the coefficients in the problem can be perturbed (in the line of [18]), are [2,[8][9][10]12,13,16], etc.…”