Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization

Abstract

In this paper, we study a class of constrained scalar set-valued optimization problems, which includes scalar optimization problems with cone constraints as special cases. We introduce (local) calmness of order α for this class of constrained scalar set-valued optimization problems. We show that the (local) calmness of order α is equivalent to the existence of a (local) exact set-valued penalty map.

About[q]-regularity properties of collections of sets

“…Then, with f (x) = x 2 , we have f ′ (z ρ ) ≤ 1 = f ′ 1 2 for any ρ ∈ 0, 1 2 . Thus, the lines h ρ and l ρ through x ρ and z ρ , respectively, with the slope f ′ (z ρ ) separate the constraint set in (22) and Ω 1 and consequently, for any x in the constraint set in (22), it holds…”