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## Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2002 | 22 | 1 | 33-66
Tytuł artykułu

### Approximation of set-valued functions by single-valued one

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Treść / Zawartość
Warianty tytułu
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
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
33-66
Opis fizyczny
Daty
wydano
2002
otrzymano
2002-01-25
Twórcy
autor
• Technical University of Varna, BG-9010 Varna, Bulgaria
autor
• Technical University of Ilmenau, D-98684 Ilmenau, PF 100565, Germany
Bibliografia
• [1] J.P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin, Heidelberg, New York, Tokyo 1984.
• [2] P.K. Belobrov, K voprosu o chebyshevskom sentre mnozhestva, Izvestija vysshich uchebnych zavedenij 38 (1) (1964), 3-9. (in Russian)
• [3] B. Bank, J. Guddat, D. Klatte, K. Kummer, K. Tammer, Non-Linear Parametric Optimization, Akademie Verlag Berlin 1982.
• [4] F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, New York 1983.
• [5] V.F. Demjanov and A.M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt am Main 1995.
• [6] A.L. Garkavi, On the best net and the best cut of a set in a normed space, Izv. Akad. Nauk SSSR, Ser. Mat. 26 (1962), 87-106. (in Russian)
• [7] A.L. Garkavi, On the Chebyshev center and the convex hull of a set, Usp. Mat. Nauk 19 (6) (1964), 139-145, (120). (in Russian)
• [8] A.L. Garkavi, Minimax balayge theorem and an inscribed ball problem, Matematicheskie Zametki 30 (1) (1981), 109-121. (in Russian)
• [9] I. Ginchev and A. Hoffmann, On the best approximation of set-valued functions, in: P. Gritzmann, R. Horst, E. Sachs, R. Tichatschke (eds.), Recent Advances in Optimization (Proc. of the 8th French-German Conference on Optimization, Trier, July 21-26, 1996, Lect. Notes Econ. Math. Syst. 452, Springer, Berlin Heidelberg 1997, 61-74.
• [10] P.M. Gruber, The space of convex bodies, in: P.M. Gruber, J.M. Wills (eds.), Handbook of Convex Geometry, Volume A, North-Holland, Amsterdam 1993, 301-318.
• [11] R. Hettich and P. Zencke, Numerische Methoden der Approximation und Semi-Infiniten Optimierung, B.G. Teubner, Leipzig 1982.
• [12] A. Hoffmann, The distance to the intersection of two convex sets expressed by the distances to each of them, Math. Nachr. 157 (1992), 81-98.
• [13] J.B. Hiriart Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer Verlag, Berlin 1993.
• [14] R.B. Holmes, Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics 24, Springer, New York-Heidelberg-Berlin 1975.
• [15] R. Horst and H. Tuy, Global Optimization, Deterministic Approaches, Springer, Berlin etc. 1990.
• [16] P. Kosmol, Optimierung und Approximation, Walter de Gruyter, Berlin 1991.
• [17] P.J. Laurent, Approximation et Optimisation, Enseignement des Sciences 13, Hermann, Paris 1972.
• [18] K. Leichtweiss, Konvexe Mengen, Deutscher Verlag der Wissenschaften, Berlin 1980.
• [19] L.E. Rybiński, Continuous Selections and Variational Systems, Monografie 61, Institute of Mathematics, Higher College of Engineering, Zielona Góra, Poland 1992, 100 pp, ISSN 0239-7390.
• [20] R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Grundlehren der mathematischen Wissenschaften 317, Springer, Berlin 1998.
• [21] I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Grundlehren der mathematischen Wissenschaften 171, Springer, Berlin 1970.
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Bibliografia
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