Approximation of set-valued functions by single-valued one

• Autorzy: Ivan Ginchev , Armin Hoffmann
• Języki publikacji: EN

Abstrakty

EN

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 single-valued function g from a given closed convex set V ⊂ C(M,Y). In an abstract setting this problem is posed as the extremal problem $sup_{t ∈ M} ρ(g(t), (t)) → inf$, 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.

Słowa kluczowe

EN

Chebyshev approximation; set-valued functions; convex optimization

Kategorie tematyczne

• 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
• 49J53: Set-valued and variational analysis
• 90C25: Convex programming
• 41A50: Best approximation, Chebyshev systems

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae, Differential Inclusions, Control and Optimization
• Rocznik: 2002
• Tom: 22
• Numer: 1
• Strony: 33-66

Daty

wydano

2002

otrzymano

2002-01-25

Twórcy

autor

Ivan Ginchev

• Technical University of Varna, BG-9010 Varna, Bulgaria

autor

Armin Hoffmann

• Technical University of Ilmenau, D-98684 Ilmenau, PF 100565, Germany

Bibliografia

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Identyfikatory

DOI

10.7151/dmgt.1031