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Newton’s method for convex programming and Tschebyscheff approximation. (English) Zbl 0113.10703


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[1] Remez, E.: Sur un, Procédé Convergent d’Approximations Successives pour Determiner les Polynomes d’Appproximation. C. R. Acad. Sci. Paris198, 2063-2065 (1934). · JFM 60.0211.03
[2] Remez, E.: Sur le Calcul Effectif des Polynomes d’Approximation deTschebyscheff. C. R. Acad. Sci. Paris199, 337-340 (1934). · JFM 60.0212.01
[3] Remez, E. Ya.: On the Method of Best, in the Sense ofTchebycheff, Approximate Representation of Functions, (Ukrainian), Kiev, 1935. See also Reference 4 below.
[4] Remez, E. Ya. General Computation Methods for Chebyshev Approximation. Problems with Real Parameters Entering Linearly. Izdat. Akad. Nauk Ukrainsk. SSR. Kiev, 1957. 454 pp. See also MR 19-580 (Russian).
[5] Novodvorskii, E. N., andI. Sh. Pinsker: On a Process of Equalization of Maxima, Uspehi Matem. Nauk, N. S.6, 174-181 (1951). See alsoShenitzer, A.: Chebyshev Approximations. J. Assoc. Comput. Mach.4, 30-35 (1957). MR 13-728.
[6] Beale, E. M. L.: An Alternative Method for Linear Programming. Proc. Cambridge Philos. Soc.50, 512-523 (1954). MR 16-155. · Zbl 0056.13705
[7] Beale, E. M. L.: On Minimizing, a Convex Function Subject to Linear Inequalities. J. Roy. Stat. Soc., Ser. B17, 173-177 (1955). · Zbl 0068.13701
[8] Bratton, Donald: New Results in the Theory and Techniques of Chebyshev Fitting. Abstract 546-34. Notices Am. Math. Soc.5, 248 (1958).
[9] Stiefel, Eduard L.: Numerical Methods of Tchebycheff Approximation, pp. 217-232 inR. E. Langer (ed.), On Numerical Approximation. Madison 1959. 480 pp. · Zbl 0083.35502
[10] Stiefel, E.: Über diskrete und lineare Tschebyscheff-Approximationen. Numerische Mathematik1, 1-28 (1959). · Zbl 0083.11501
[11] Wolfe, Philip: Programming with Nonlinear Constraints. Preliminary Report, Abstract 548-102. Notices Am. Math. Soc.5, 508 (1958).
[12] Stone, Jeremy J. The Cross Section Method, presented orally at Symposium for Mathematical Programming, RAND Corporation, March 19, 1959.
[13] Kelley, James E.: A General Technique for Convex Programming, presented orally at Symposium on Mathematical Programming, RAND Corporation, March 19, 1959.
[14] Cheney, E. W., andA. A. Goldstein: Proximity Maps for Convex Sets. Proc. Amer. Math. Soc.10, 448-450 (1959). · Zbl 0092.11403
[15] Bonnesen, T., u.W. Fenchel: Theorie der konvexen Körper. Berlin 1934. · Zbl 0008.07708
[16] Fan, K.: On Systems of Linear Inequalities. In: Linear Inequalities and Related Systems, ed., byH. W. Kuhn andA. W. Tucker, pp. 99-156. Princeton 1956. · Zbl 0072.37602
[17] Bram, Joseph: Chebychev Approximation in Locally Compact Spaces. Proc. Am. Math. Soc.9, 133-136 (1958). · Zbl 0080.04402
[18] Goldstein, A. A., andE. W. Cheney: A Finite Algorithm for the Solution of Consistent Linear Equations and Inequalities and for the Tchebycheff Approximation of Inconsistent Linear Equations. Pac. J. Math.8, 415-427 (1958). · Zbl 0084.01902
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