Chebyshev approximations do not need the Haar condition

V. I. Zorkal'tsev

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 196 (2021), pp. 28-35.

Résumé

In this paper, we consider the problem of constructing a Chebyshev projection of the coordinate origin onto a linear manifold. In particular, the Chebyshev linear approximation problem can be formulated in this form. We present an algorithm for determining Chebyshev projections, which is not based on the Haar condition. The algorithm consists of finding relatively interior points of optimal solutions of a finite sequence of linear programming problems.

Détail

Keywords: Chebyshev projection, Hölder projection, linear approximation.
Mots-clés : Haar condition, optimal solution

Bibliographie

[1] Aganbegyan A. G., Granberg A. G., Ekonomiko-matematicheskii analiz mezhotraslevogo balansa SSSR, Mysl, M., 1968

[2] Vatolin A. A., “Ob approksimatsii nesovmestnykh sistem lineinykh uravnenii i neravenstv”, Metody approksimatsii nesobstvennykh zadach matematicheskogo programmirovaniya, Sverdlovsk, 1984, 20–25

[3] Demyanov V. F., Malozemov V. N., Vvedenie v minimaks, Nauka, M., 1972 | MR

[4] Dikin I. I., Zorkaltsev V. I., Iterativnoe reshenie zadach matematicheskogo programmirovaniya (algoritmy metoda vnutrennikh tochek), Nauka, Novosibirsk, 1980 | MR

[5] Eremin I. I., Teoriya lineinoi optimizatsii, Ekaterinburg, 1999

[6] Zorkaltsev V. I., Batoeva I. V., Bedenkov A. R., Sadov S. L., Soglasovanie chastnykh prognozov v balansovykh modelyakh, Syktyvkar, 1990

[7] Zorkaltsev V. I., Problemy agregirovaniya v ekonomike: est li logicheskaya sovmestimost mikroekonomiki i makroekonomiki?, Irkutsk, 1997

[8] Zorkaltsev V. I., Chernikova L. I., Renta, nalogi i struktura tsen, Irkutsk, 1991

[9] Zorkaltsev V. I., Metod naimenshikh kvadratov i ego konkurenty, Irkutsk, 1993

[10] Zorkaltsev V. I., Metod naimenshikh kvadratov: geometricheskie svoistva, alternativnye podkhody, prilozheniya, Nauka, Novosibirsk, 1995 | MR

[11] Zorkaltsev V. I., “Naimenee udalennye ot nachala koordinat tochki lineinogo mnogoobraziya”, Zh. vychisl. mat. mat. fiz., 35:5 (1995), 801–810 | MR | Zbl

[12] Zorkaltsev V. I., “Naimenee udalennye ot nachala koordinat resheniya sistemy lineinykh neravenstv”, Izv. Irkutsk. un-ta. Ser. Mat., 4:2 (2011), 102–103

[13] Zorkaltsev V. I., “Oktaedricheskie i evklidovy proektsii tochki na lineinoe mnogoobrazie”, Tr. In-ta mat. mekh. UrO RAN., 18:3 (2011), 106–118

[14] Zorkaltsev V. I., “Proektsii tochki na poliedr”, Zh. vychisl. mat. mat. fiz., 53:1 (2013), 4–19 | Zbl

[15] Zorkaltsev V. I., Kiseleva M. A., Sistemy lineinykh neravenstv, Irkutsk, 2007

[16] Kollatts A., Krabe V., Teoriya priblizhenii. Chebyshevskie priblizheniya, Nauka, M., 1978

[17] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968 | MR

[18] Louson Ch., Khenson R., Chislennoe reshenie zadach metoda naimenshikh kvadratov, Nauka, M., 1986 | MR

[19] Samarskii A. A., Gulin A. V., Chislennye metody, Nauka, M., 1989

[20] Cherkassovskii B. V., “Zadachi balansirovki matrits”, Metody matematicheskogo programmirovaniya i programmnoe obespechenie, UrO AN, Sverdlovsk, 1984, 216–217

[21] Haare A., “Die Minkowskische Geometrie und die Annaherung an stetige Funktionen”, Math. Ann., 78:1–4 (1917), 294–311 | DOI | MR