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Some Theoretical Aspects of Newton’s Method for Constrained Best Interpolation

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Continuous Optimization

Part of the book series: Applied Optimization ((APOP,volume 99))

Summary

The paper contains new results as well as surveys on recent developments on the constrained best interpolation problem, and in particular on the convex best interpolation problem. Issues addressed include theoretical reduction of the problem to a system of nonsmooth equations, nonsmooth analysis of those equations and development of Newton’s method, convergence analysis and globalization. We frequently use the convex best interpolation to illustrate the seemingly complex theory. Important techniques such as splitting are introduced and interesting links between approaches from approximation and optimization are also established. Open problems related to polyhedral constraints and strips may be tackled by the tools introduced and developed in this paper.

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References

  1. Andersson, L.-E., Elfving, T.: An algorithm for constrained interpolation. SIAM J. Sci. Statist. Comput., 8, 1012–1025 (1987)

    Article  MathSciNet  Google Scholar 

  2. Andersson, L.-E., Elfving, T.: Best constrained approximation in Hilbert space and interpolation by cubic splines subject to obstacles. SIAM J. Sci. Comput., 16, 1209–1232 (1995)

    Article  MathSciNet  Google Scholar 

  3. Bauschke, H.H., Borwein, J.M., Tseng, P.: Bounded linear regularity, strong CHIP, and CHIP are distinct properties. J. Convex Anal., 7, 395–412 (2000)

    MathSciNet  Google Scholar 

  4. Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program., 86, 135–160 (1999)

    Article  MathSciNet  Google Scholar 

  5. Borwein, J., Lewis, A.S.: Partially finite convex programming I: Quasi relative interiors and duality theory. Math. Program. 57, 15–48 (1992)

    Article  MathSciNet  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley & Sons, New York (1983)

    Google Scholar 

  7. Chui, C.K., Deutsch, F., Ward, J.D.: Constrained best approximation in Hilbert space. Constr. Approx., 6, 35–64 (1990)

    Article  MathSciNet  Google Scholar 

  8. Chui, C.K., Deutsch, F., Ward, J.D.: Constrained best approximation in Hilbert space II, J. Approx. Theory, 71 (1992), pp. 213–238.

    Article  MathSciNet  Google Scholar 

  9. de Boor, C: A Practical Guide to Splines. Springer-Verlag, New York (1978)

    Google Scholar 

  10. De Luca, T., Facchinei, F., Kanzow, C: A semismooth equation approach to the solution of nonlinear complementarity problems. Math. Program., 75, 407–439 (1996)

    Article  Google Scholar 

  11. Deutsch, F.: Best approximation in inner product spaces. CMS Books in Mathematics 7. Springer-Verlag, New York (2001)

    Google Scholar 

  12. Deutsch, F., Li, W., Ward, J.D.: A dual approach to constrained interpolation from a convex subset of Hilbert space. J. Approx. Theory, 90, 385–414 (1997)

    Article  MathSciNet  Google Scholar 

  13. Deutsch, F., Li, W., Ward, J.D.: Best approximation from the intersection of a closed convex set and a polyhedron in Hilbert space, weak Slater conditions, and the strong conical hull intersection property. SIAM J. Optim., 10, 252–268 (1999)

    Article  MathSciNet  Google Scholar 

  14. Deutsch, F., Ubhaya, V.A., Ward, J.D., Xu, Y.: Constrained best approximation in Hilbert space. III. Applications to n-convex functions. Constr. Approx., 12, 361–384 (1996)

    Article  MathSciNet  Google Scholar 

  15. Dontchev, A.L.: Best interpolation in a strip. J. Approx. Theory, 73 334–342 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  16. Dontchev, A.L., Kalchev, B.D.: Duality and well-posedness in convex interpolation. Numer. Funct. Anal, and Optim., 10, 673–689 (1989)

    MathSciNet  Google Scholar 

  17. Dontchev, A.L., Kolmanovsky, I.: Best interpolation in a strip. II. Reduction to unconstrained convex optimization. Comput. Optim. Appl., 5, 233–251 (1996)

    Article  MathSciNet  Google Scholar 

  18. Dontchev, A.L., Qi, H.-D., Qi, L.: Convergence of Newton’s method for convex best interpolation. Numer. Math., 87 435–456 (2001)

    Article  MathSciNet  Google Scholar 

  19. Dontchev, A.L., Qi, H.-D., Qi, L.: Quadratic convergence of Newton’s method for convex interpolation and smoothing. Constr. Approx., 19, 123–143 (2003)

    Article  MathSciNet  Google Scholar 

  20. Dontchev, A.L., Qi, H.-D., Qi, L., Yin, H.: A Newton method for shape-preserving spline interpolation. SIAM J. Optim., 13, 588–602 (2002)

    Article  MathSciNet  Google Scholar 

  21. Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems, Vol. I & II. Springer-Verlag, New York (2003)

    Google Scholar 

  22. Favard, J.: Sur l’interpolation. J. Math. Pures Appl., 19, 281–306 (1940)

    MATH  MathSciNet  Google Scholar 

  23. Fischer, A.: A special Newton-type optimization method. Optimization, 24, 269–284 (1992)

    MATH  MathSciNet  Google Scholar 

  24. Gowda, M.S., Teboulle, M.: A comparison of constraint qualifications in infinite-dimensional convex programming. SIAM J. Control Optim., 28, 925–935 (1990)

    Article  MathSciNet  Google Scholar 

  25. Hornung, U.: Interpolation by smooth functions under restriction on the derivatives. J. Approx. Theory, 28, 227–237 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  26. Iliev, G., Pollul, W.: Convex interpolation by functions with minimal Lp norm (1 < p < ∞) of the kth derivative. Mathematics and mathematical education (Sunny Beach, 1984), 31–42, Bulg. Akad. Nauk, Sofia (1984)

    Google Scholar 

  27. Irvine, L.D., Marin, S.P., Smith, P.W.: Constrained interpolation and smoothing. Constr. Approx., 2, 129–151 (1986)

    Article  MathSciNet  Google Scholar 

  28. V. Jeyakumar: Infinite-dimensional convex programming with applications to constrained approximation. J. Optim. Theory Appl., 75, 569–586 (1992)

    Article  MathSciNet  Google Scholar 

  29. V. Jeyakumar, D.T. Luc: Approximate Jacobian matrices for nonsmooth continuous maps and C1-optimization. SIAM J. Control Optim., 36, 1815–1832 (1998)

    Article  MathSciNet  Google Scholar 

  30. V. Jeyakumar, H. Wolkowicz: Generalizations of Slater’s constraint qualification for infinite convex programs. Math. Program., 57, 85–101 (1992)

    Article  MathSciNet  Google Scholar 

  31. Jiang, H., Qi, L.: Local uniqueness and Newton-type methods for non-smooth variational inequalities. J. Math. Analysis and Appl., 196 314–331 (1995)

    Article  MathSciNet  Google Scholar 

  32. Klatte D., Kummer, B.: Nonsmooth equations in optimization. Regularity, calculus, methods and applications. Nonconvex Optimization and its Applications, 60. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  33. B. Kummer: Newton’s method for nondifferentiable functions. Advances in mathematical optimization, 114–125, Math. Res., 45, Akademie-Verlag, Berlin (1988)

    Google Scholar 

  34. Li, C, Jin, X.Q.: Nonlinearly constrained best approximation in Hilbert spaces: the strong chip and the basic constraint qualification. SIAM J. Optim., 13, 228–239 (2002)

    Article  MathSciNet  Google Scholar 

  35. Li, C., Ng, K.F.: On best approximation by nonconvex sets and perturbation of nonconvex inequality systems in Hilbert spaces. SIAM J. Optim., 13, 726–744 (2002)

    Article  MathSciNet  Google Scholar 

  36. Li, C., Ng, K.F.: Constraint qualification, the strong chip, and best approximation with convex constraints in Banach spaces. SIAM J. Optim., 14, 584–607 (2003)

    Article  MathSciNet  Google Scholar 

  37. Micchelli, C.A., Smith, P.W., Swetits, J., Ward, J.D.: Constrained Lp approximation. Constr. Approx., 1, 93–102 (1985)

    Article  MathSciNet  Google Scholar 

  38. Micchelli, C.A., Utreras, F.I.: Smoothing and interpolation in a convex subset of a Hilbert space. SIAM J. Sci. Statist. Comput., 9, 728–747 (1988)

    Article  MathSciNet  Google Scholar 

  39. Mifflin, R.: Semismoothness and semiconvex functions in constrained optimization. SIAM J. Control Optim., 15, 959–972 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  40. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program., 58, 353–367 (1993)

    Article  MathSciNet  Google Scholar 

  41. Ralph, D.: Personal communication. May. (2002)

    Google Scholar 

  42. Sun, D., Qi, L.: On NCP-functions. Comput. Optim. Appl., 13, 201–220 (1999)

    Article  MathSciNet  Google Scholar 

  43. Xu, H.: Set-valued approximations and Newton’s methods. Math. Program., 84, 401–420 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, The University of Southampton Highfield, Southampton SO17 1BJ, UK

    Hou-Duo Qi

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  1. Hou-Duo Qi
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Editors and Affiliations

  1. University of New South Wales, Sydney, Australia

    Vaithilingam Jeyakumar

  2. University of Ballarat, Ballarat, Australia

    Alexander Rubinov

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Qi, HD. (2005). Some Theoretical Aspects of Newton’s Method for Constrained Best Interpolation. In: Jeyakumar, V., Rubinov, A. (eds) Continuous Optimization. Applied Optimization, vol 99. Springer, Boston, MA. https://doi.org/10.1007/0-387-26771-9_2

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