On stability and newton-type methods for lipschitzian equations with applications to optimization problems

  • Bernd Kummer
I Optimality And Duality
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)


Locally Lipschitz Inverse and implicit function Newton’s Method Generalized derivatives Multifunctions Convergence analysis Approximate solutions Critical points in optimization 


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  1. [1]
    Aubin, J.P. & Ekeland, I. Applied Nonlinear Analysis. Wiley, New York, 1984zbMATHGoogle Scholar
  2. [2]
    Aubin, J.P. & Frankowska, H. Set-valued Analysis. Birkhaüser, Basel, 1990zbMATHGoogle Scholar
  3. [3]
    Aze, D. An inversion theorem for set-valued maps. Bull. Austral. Math. Soc. 37 (1988) pp. 411–414zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Clarke, F.H. On the inverse function theorem. Pacific Journ. Math. 64, No. 1 (1976) pp. 97–102zbMATHGoogle Scholar
  5. [5]
    Clarke, F.H. Optimization and Nonsmooth Analysis. Wiley, NewYork, 1983zbMATHGoogle Scholar
  6. [6]
    Harker, P.T. & Pang, J.-S. Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications. Mathematical Programming 48, (1990) pp. 161–220zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Ioffe, A.D. Nonsmooth analysis: Differential calculus of nondifferentiable mappings. Trans. Amer. Math. Soc. 266, (1981) pp. 1–56zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Jittorntrum, K. Solution point differentiability without strict complementarity in nonlinear programming. Math. Programming Study 21, (1984) pp. 127–138zbMATHMathSciNetGoogle Scholar
  9. [9]
    Jongen, H.Th., Klatte, D., Tammer, K. Implicit functions and sensitivity of stationary points. Preprint No. 1, Lehrstuhl C für Mathematik, RWTH Aachen, D-5100 AachenGoogle Scholar
  10. [10]
    Kojima, M. Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points, S.M. Robinson ed., Academic Press, New York, 1980 pp. 93–138Google Scholar
  11. [11]
    Kojima, M. & Shindo, S. Extensions of Newton and quasi-Newton methods to systems of PC 1 equations. Journ. of Operations Research Soc. of Japan 29 (1987) pp. 352–374MathSciNetGoogle Scholar
  12. [12]
    Kummer, B. Newton’s method for non-differentiable functions. In: Advances in Math. Optimization, J. Guddat et al. eds. Akademie Verlag Berlin, Ser. Mathem. Res. Vol 45, 1988 pp. 114–125MathSciNetGoogle Scholar
  13. [13]
    Kummer, B. Lipschitzian inverse functions, directional derivatives and application in C 1,1-optimization. Working Paper WP-89-084 (1989) IIASA Laxenburg, AustriaGoogle Scholar
  14. [14]
    Kummer, B. An implicit function theorem for C 0,1-equations and parametric C 1,1-optimization. Journ. Math. Analysis & Appl. (1991) Vol, 158, No.1, pp. 35–46zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Kummer, B. Newton’s method based on generalized derivatives for nonsmooth functions: convergence analysis. Preprint (1991) Humboldt-University, Deptm. of Appl. MathematicsGoogle Scholar
  16. [16]
    Langenbach, A. Über lipschitzstetige implizite Funktionen. Zeitschr. für Analysis und ihre Anwendungen Bd. 8 (3), (1989) pp. 289–292zbMATHMathSciNetGoogle Scholar
  17. [17]
    Mordukhovich, B.S. On sensitivity and stability analysis in nonsmooth optimization. Preprint (1991), Deptm. of Mathem., Wayne State University, Detroit, Michigan 48202, USAGoogle Scholar
  18. [18]
    Ortega J.M. & Rheinboldt W.C. Iterative Solution of Nonlinear Equations of Several Variables. Academic Press, San Diego, 1970zbMATHGoogle Scholar
  19. [19]
    Pang, J.-S. Newton’s method for B-differentiable equations. Math. of Operations Res. 15, (1990) pp. 311–341zbMATHGoogle Scholar
  20. [20]
    Pang, J.-S. & Gabriel, S.A. NE/SQP: A robust algorithm for the nonlinear complementarity problem. Working Paper, (1991), Department of Math. Sc., The Johns Hopkins Univ., Baltimore Maryland 21218Google Scholar
  21. [21]
    Park, K. Continuation methods for nonlinear programming. Ph.D.Dissertation, (1989), Department of Industrial Engineering, Univ. of Wisconsin-MadisonGoogle Scholar
  22. [22]
    Qi, L. Convergence analysis of some algorithms for solving nonsmooth equations. Manuscript, (1991), School of Math., The Univ. of New South Wales, Kensington, New South WalesGoogle Scholar
  23. [23]
    Ralph, D. Global convergence of damped Newton’s method for nonsmooth equations, via the path search. Techn. Report TR 90-1181, (1990), Department of Computer Sc., Cornell Univ. Ithaca, New YorkGoogle Scholar
  24. [24]
    Robinson, S.M. Newton’s method for a class of nonsmooth functions. Working Paper, (1988), Univ. of Wisconsin-Madison, Department of Industrial Engineering, Madison, WI 53706Google Scholar
  25. [25]
    Robinson, S.M. An implicit-function theorem for a class of nonsmooth functions. Mathematics of OR, Vol. 16, No. 2, (1991) pp. 292–309zbMATHGoogle Scholar
  26. [26]
    Thibault, L. Subdifferentials of compactly Lipschitzian vector-valued functions. Ann. Mat. Pura Appl. (4) 125, (1980) pp. 157–192zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Thibault, L. On generalized differentials and subdifferentials of Lipschitz vector-valued functions. Nonlinear Analysis Theory Methods Appl. 6 (10), (1982) pp. 1037–1053zbMATHCrossRefMathSciNetGoogle Scholar

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© International Federation for Information Processing 1992

Authors and Affiliations

  • Bernd Kummer
    • 1
  1. 1.Department of MathematicsHumboldt-University BerlinBerlin

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