Approximation of set-valued functions by single-valued one

Abstract

Let Σ : M → 2 Y \ {∅} be a set-valued function defined on a Hausdorff compact topological space M and taking values in the normed space (Y, •). We deal with the problem of finding the best Chebyshev type approximation of the set-valued function Σ by a singlevalued function g from a given closed convex set V ⊂ C(M, Y). In an abstract setting this problem is posed as the extremal problem sup t∈M ρ(g(t), Σ(t)) → inf, g ∈ V. Here ρ is a functional whose values ρ(q, S) can be interpreted as some distance from the point q to the set S ⊂ Y. In the paper, we are confined to two natural distance functionals ρ = H and ρ = D. H(q, S) is the Hausdorff distance (the excess) from the point q to the set cl S, and D(q, S) is referred to as the oriented distance from the point q to set cl conv S. We prove that both these problems are convex optimization problems. While distinguishing between the so called regular and irregular case problems, in particular the case V = C(M, Y) is studied to show that the solutions in the irregular case are obtained as continuous selections of certain set-valued maps. In the general case, optimality conditions in terms of directional derivatives are obtained of both primal and dual type.

Some remarks on the Minty vector variational principle

“…The oriented distance from y to A is given by the function D(y, A) = d A (y) − d Y \A (y). It can be shown [13] that when A is a convex cone we have D(y, A) = sup{ ξ, y | ξ ∈ C , ξ = 1}. For x * ∈ K, we consider the function φ : X → Y defined as…”

From Scalar to Vector Optimization

“…Zaffaroni [32] gives different notions of efficiency and uses the function D for their scalarization and comparison. Ginchev, Hoffmann [14] use the oriented distance to study approximation of set-valued functions by single-valued ones and in the case of a convex set A show the representation D(y, A) = sup ξ =1…”

On scalar and vector -stable functions

“…It is known [24] that when M ⊂ R k is a closed convex cone, then D(y, −M) = sup ξ ∈Γ M ′ ⟨ξ , y⟩. It seems that the oriented distance, known also as the signed distance, was first introduced by Hiriart-Urruty [25].…”