Duality for location problems with unbounded unit balls

A sequential formula for the $\varepsilon$-subdifferential of multi-composed functions via a perturbation approach with an application to location problems

“…Motivated by the location models examined in [17,3], the aim of this section is to give the sequential ε-optimality conditions of the following location problem with geometric constraint…”

“…Since the functions j + C 0 and f 1 , • • • , f q are defined in a general way, problem (L P) covers a large class in location problems, such as the Weber problem with gauges of closed convex sets [17], single minimax location problems with or without set-up costs [6] and so on. In order to justify this point, we consider, for instance, the Weber problem with gauges of closed convex sets studied in [17]…”

“…It is worth noting that the defined functions j + C 0 and j C 1 , • • • , j C q are convex and continuous with dom j + C 0 = R q and dom j C 1 = • • • = dom j C q = X (see [17,Proposition 4.1] and [6, Theorem 1 and Remark 2]). Thus problem (L P) is a convex optimization problem.…”

“…Since f i : R → R is proper, convex, lower semicontinuous, and nondecreasing on dom f i = R + , i = 1, • • • , q, it follows that h 1 is proper, R q + -convex, (R q + , R q + )nondecreasing on domh 1 = R q + , and R q + -epi closed with h 1 (domh 1 ) ⊆ R q + . As regards j + C 0 , it is proper, convex, R q + -nondecreasing on dom j + C 0 = R q , and lower semicontinuous (see [17,Proposition 4.1]). Hence, the functions f := δ C , ϕ ≡ 0, ψ ≡ 0, g := j + C 0 , h 1 , and h 2 verify all the conditions considered in Theorem 4.1.…”