Applications of Analytic Centers

  • György Sonnevend
Part of the NATO ASI Series book series (volume 70)

Abstract

In this paper we present some applications of a recently introduced concept, the ”analytical center” of a convex inequality system. Let F = {f a (x, p)} denote a family of functions depending on parameters pP and elements a of an index set A, which are concave in xX, on the sets S a,p where they are nonnegative. Here X is a Hilbert space fixed together with the sets A and P. Below we shall consider the specific case when for each (a, p) the function f a (x, p) is linear or concave quadratic in x or when it is the determinant of a symmetric, positive semidefinite matrix, whose elements depend linearly on x.

Keywords

Entropy Assure Tral Geophysics 

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References

  1. [1]
    D.A. Bayer and Lagarias, The nonlinear geometry of linear programming, I, II,III, preprints, AT&T Bell Laboratories, Murray Hill, New Jersey, 1986–7.Google Scholar
  2. [2]
    J.P. Burg, Maximum entropy spectral analysis, in Modern Spectrum Analysis (D.G. Childiers ed.), IEEE Press, New York, 1978, 34–39.Google Scholar
  3. [3]
    T. Constantinescu et al., Schur analysis of some completion problems, Preprint N. 62, INCREST, Bucharest, 1986.Google Scholar
  4. [4]
    Ch. Davies et al.,Norm preserving dilatations and their applications to optimal error bounds, SIAM J. Num. Anal., 19 (1983) 445–469.CrossRefGoogle Scholar
  5. [5]
    M. A. Dahleh, J.B. Pearson, Optimal Rejection of Bounded Disturbances, IEEE Trans. Aut. Contr., 33 (1988) 722–731.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    P. Dewilde, E.F.A. Deprettere, The generalized Schur algorithm: Approximation and Hierarchy, in Operator Theory Advances and Applications, Birkhäuser (1988) in press.Google Scholar
  7. [7]
    H. Dym, J- contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation, preprint 1988, Dept. Theor. Math., The Weizman Inst., Rehovot.Google Scholar
  8. [8]
    B. Fritzsche, B. Kirstein, A matrix extension problem with entropy optimization, Optimization 19 (1988) 85–99.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    R. Grone et al., Positive definite completions of partial Hermitian matrices, Linear Algebra and its Applications 58 (1984) 109–124.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    F. Jarre: On the Convergence of the Method of Analytic Centers when applied to Convex Quadratic Programs, report No. 35, 1987, Schwerpunktprogramm der DFG für anwendungsbezogene Optimierung und Steuerung, submitted to Math. Programming in revised form, March 1988.Google Scholar
  11. F. Jarre: Convergence of the Method of Analytic Centers for Generalized Conver Programs, DFG - report No.67, March 1988.Google Scholar
  12. [11]
    H.J. Landau, Maximum enzropy and the moment problem, Bull. Amer. Math. Soc. 16 (1987) 47–77.MathSciNetMATHCrossRefGoogle Scholar
  13. [12]
    Programming, Proc. 12’ IFIP Conf. on System Modelling and Optimization, Budapest 1985, Lecture Notes in Control and Inf. Sciences 84, 866–876, Springer Verlag 1986Google Scholar
  14. [13]
    G. Sonnevend, Sequential and stable methods for recovering the reflectivity function of a layered medium, in Model Optimization in Exploration Geophysics, 2, (ed. A. Vogel),1987, Fr. Vieveg & Sohn, 60–72.Google Scholar
  15. [14]
    G. Sonnevend, Existence and numerical computation of extremal invariant sets in linear differential games, Lect.Notes in Contr. and Inf. Sci. 22 (1981) 251–260.MathSciNetCrossRefGoogle Scholar
  16. [15]
    G. Sonnevend and J. Stoer: Global Ellipsoidal Approximations and Homotopy Methods for Solving Convex Analytic Programs, Report No. 40, Jan. 1988, Schwerpunktprogramm der Deutschen Forschungsgemeinschaft - Anwendungsbezogene Optimierung und Steuerung, Inst. f. Ang. Mathematik und Statistik, Universität Würzburg, to appear in Applied Mathematics and OptimizationGoogle Scholar
  17. [16]
    B. Szökefalvi-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North Holland, Amsterdam, 1970.Google Scholar
  18. [17]
    L.N. Trefethen, M. Gutknecht, Pade, Stable Pade and Chebyshev approximation, in Numerical Analysis (D.F. Griffiths and G.A. Watson eds. ) 1986.Google Scholar
  19. [18]
    P. Vaidya, A locally well behaved potential function and a simple Newton method to find the center of a polytope, preprint AT&T Bell Laboratories, Murray Hill, New Jersey, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • György Sonnevend
    • 1
  1. 1.Institut für Angewandte Mathematik und StatistikUniversität WürzburgWürzburgWest-Germany

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