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Articles

The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions

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Pages 999-1019 | Received 16 Nov 2020, Accepted 26 May 2021, Published online: 15 Jun 2021

References

  • Northrop RB. Signals and systems analysis in biomedical engineering. Boca Raton (FL): CRC Press; 2003.
  • Nürnberger G. Approximation by spline functions. New York (NY): Springer-Verlag; 1989.
  • Schumaker L. Uniform approximation by Chebyshev spline functions. II: free knots. SIAM J Numer Anal. 1968;5:647–656.
  • Sukhorukova N. Uniform approximation by the highest defect continuous polynomial splines: necessary and sufficient optimality conditions and their generalisations. J Optim Theory Appl. 2010;147(2):378–394.
  • Crouzeix JP, Sukhorukova N, Ugon J. Finite alternation theorems and a constructive approach to piecewise polynomial approximation in Chebyshev norm. Set-valued Var Anal. 2020;28(1):123–147.
  • Meinardus G, Nürnberger G, Sommer M, et al. Algorithms for piecewise polynomials and splines with free knots. Math Comput. 1989;53:235–247.
  • Mulansky B. Chebyshev approximation by spline functions with free knots. IMA J Numer Anal. 1992;12:95–105.
  • Sukhorukova N, Ugon J. Characterisation theorem for best polynomial spline approximation with free knots. Trans Amer Math Soc. 2017;369(9):6389–6405.
  • Sukhorukova N, Ugon J. Characterization theorem for best linear spline approximation with free knots. Dyn Contin Discrete Impuls Syst. 2010;17(5):687–708.
  • Borwein P, Daubechies I, Totik V, et al. Bivariate segment approximation and free knot splines: research problems 96-4. Constr Approx. 1996;12(4):555–558.
  • Achieser NI. Theory of approximation. New York (NY): Frederick Ungar; 1965.
  • Boehm B. Functions whose best rational Chebyshev approximation are polynomials. Numer Math. 1964;6(1):235–242.
  • Meinardus G. Approximation of functions: theory and numerical methods. Berlin: Springer-Verlag; 1967.
  • Ralston A. Rational Chebyshev approximation by Remes' algorithms. Numer Math. 1965;7(4):322–330.
  • Rivlin T. Polynomials of best uniform approximation to certain rational functions. Numer Math. 1962;4(1):345–349.
  • Lorentz GG, von Golitschek M, Makovoz Y. Constructive approximation: advanced problems. Vol. 304. Berlin: Springer; 1996.
  • Petrushev P, Popov V. Rational approximation of real functions. New York (NY): Cambridge University Press; 1987.
  • Fraser W, Hart J. On the computation of rational approximations to continuous functions. Commun ACM. 1962;5(7):401–403.
  • Ralston A, Rabinowitz P. A first course in numerical analysis. New York (NY): Courier Corporation; 2001.
  • Rice JR. The approximation of functions. Vol. 2: nonlinear and multivariate theory. Reading (MA), Addison-Wesley; 1969. (Addison-Wesley Series in Computer Science and Information Processing).
  • Barrodale I, Powell M, Roberts F. The differential correction algorithm for rational l∞-approximation. SIAM J Numer Anal. 1972;9(3):493–504.
  • Loeb HL. On rational fraction approximations at discrete points. Convair Astronautics, Math. Preprint No. 9; 1957.
  • Loeb HL. Algorithms for Chebyshev approximations using the ratio of linear forms. J Soc Ind Appl Math. 1960;8(3):458–465.
  • Osborne MR, Watson GA. An algorithm for minimax approximation in the nonlinear case. Comput J. 1969;12(1):63–68.
  • Nakatsukasa Y, Sete O, Trefethen LN. The AAA algorithm for rational approximation. SIAM J Sci Comput. 2018;40(3):A1494–A1522.
  • Cheney EW, Loeb HL. Generalized rational approximation. J Soc Ind Appl Math, Ser B: Numer Anal. 1964;1(1):11–25.
  • Barrodale I. Best rational approximation and strict quasi-convexity. SIAM J Numer Anal. 1973;10(1):8–12.
  • de Finetti B. Sulle stratificazioni convesse. Ann Mat Pura Appl. 1949;30(1):173–183.
  • Fenchel W, Blackett DW. Convex cones, sets, and functions. Princeton (NJ): Princeton University, Department of Mathematics, Logistics Research Project; 1953.
  • Crouzeix JP. Conditions for convexity of quasiconvex functions. Math Oper Res. 1980;5(1):120–125.
  • Daniilidis A, Hadjisavvas N, Martinez-Legaz JE. An appropriate subdifferential for quasiconvex functions. SIAM J Optim. 2002;12:407–420.
  • Dutta J, Rubinov AM. Abstract convexity. In: Hadjisavvas N, Komlósi S, Schaible SS, editors. Handbook of generalized convexity and generalized monotonicity. Vol. 76. New York (NY): Springer-Verlag; 2005. p. 293–333.
  • Rubinov AM, Simsek B. Conjugate quasiconvex nonnegative functions. Optimization. 1995;35(1):1–22.
  • Rubinov AM. Abstract convexity and global optimization. New York (NY): Kluwer Academic Publishers; 2000.
  • Boyd S, Vandenberghe L. Convex optimization. New York (NY): Cambridge University Press; 2010.
  • Martínez-Legaz JE. Quasiconvex duality theory by generalized conjugation methods. Optimization. 1988;19(5):603–652.
  • Neto JDC, Lopes JO, Travaglia MV. Algorithms for quasiconvex minimization. Optimization. 2011;60(8–9):1105–1117.
  • Bauschke HH, Lewis AS. Dykstra's algorithm with Bregman projections: a convergence proof. Optimization. 2000;48(4):409–427.
  • Cegielski A, Nimana N. Extrapolated cyclic subgradient projection methods for the convex feasibility problems and their numerical behaviour. Optimization. 2019;68(1):145–161.
  • Matsushita S, Xu L. On the finite termination of the Douglas-Rachford method for the convex feasibility problem. Optimization. 2016;65(11):2037–2047.
  • Nimana N, Farajzadeh AP, Petrot N. Adaptive subgradient method for the split quasi-convex feasibility problems. Optimization. 2016;65(10):1885–1898.
  • Yang Y, Yang Q. Some modified relaxed alternating projection methods for solving the two-sets convex feasibility problem. Optimization. 2013;62(4):509–525.
  • Zaslavski AJ. Subgradient projection algorithms and approximate solutions of convex feasibility problems. J Optim Theory Appl. 2013;57(3):803–819.
  • Zhao X, Köbis MA. On the convergence of general projection methods for solving convex feasibility problems with applications to the inverse problem of image recovery. Optimization. 2018;67(9):1409–1427.
  • Peiris V, Sharon N, Sukhorukova N, et al. Generalised rational approximation and its application to improve deep learning classifiers. Appl Math Comput. 2021;389:125560.
  • Goberna MA, López MA. A comprehensive survey of linear semi-infinite optimization theory. In: Reemtsen R, Rückmann JJ, editors. Semi-infinite programming. Nonconvex Optimization and Its Applications, Vol. 25. Boston (MA): Springer; 1998. p. 3–27.

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