Approximation of Set-valued Functions by Single-valued One

Let Σ : M → 2 \ {∅} 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 supt∈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.