Mathematical Programming

, Volume 117, Issue 1–2, pp 305–330 | Cite as

Optimization methods and stability of inclusions in Banach spaces

  • Diethard Klatte
  • Bernd Kummer


Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.


Generalized equation Variational inequality Perturbation Regularization Stability criteria Metric regularity Calmness Approximate projections Penalization Successive approximation Newton’s method 

Mathematics Subject Classification (2000)

49J52 49K40 90C31 65Y20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aubin J.-P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley, New York zbMATHGoogle Scholar
  2. 2.
    Bank B., Guddat J., Klatte D., Kummer B. and Tammer K. (1982). Non-Linear Parametric Optimization. Akademie-Verlag, Berlin Google Scholar
  3. 3.
    Bonnans J.F. and Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York zbMATHGoogle Scholar
  4. 4.
    Burke J.V. (1991). Calmness and exact penalization. SIAM J. Control Optim. 29: 493–497 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Burke, J.V., Deng, S.: Weak sharp minima revisited, Part III: Error bounds for differentiable convex inclusions. Math. Programm. Published online (2007)Google Scholar
  6. 6.
    Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York zbMATHGoogle Scholar
  7. 7.
    Cominetti R. (1990). Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optimiz. 21: 265–287 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dempe S. (2002). Foundations of Bilevel Programming. Kluwer, Dordrecht zbMATHGoogle Scholar
  9. 9.
    Dolecki S. and Rolewicz S. (1979). Exact penalties for local minima. SIAM J. Control Optim. 17: 596–606 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Dontchev A. (1996). Local convergence of the Newton method for generalized equations. Comptes Rendus de l’Acédemie des Sciences de Paris 332: 327–331 MathSciNetGoogle Scholar
  11. 11.
    Dontchev A. and Rockafellar R.T. (2004). Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12: 79–109 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ekeland I. (1974). On the variational principle. J. Math. Anal. Appl. 47: 324–353 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Facchinei F. and Pang J.-S. (2003). Finite-Dimensional Variational Inequalities and Complementary Problems, Vol I and Vol II. Springer, New York Google Scholar
  14. 14.
    Fusek P. (2001). Isolated zeros of Lipschitzian metrically regular Rn functions. Optimization 49: 425–446 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Graves L.M. (1950). Some mapping theorems. Duke Math. J. 17: 11–114 CrossRefMathSciNetGoogle Scholar
  16. 16.
    Grossmann C., Klatte D. and Kummer B. (2004). Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ. Kybernetika 40: 571–584 MathSciNetGoogle Scholar
  17. 17.
    Henrion R. and Outrata J. (2001). A subdifferential condition for calmness of multifunctions. J. Math. Anal. Appl. 258: 110–130 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Henrion R. and Outrata R. (2005). Calmness of constraint systems with applications. Math. Program. Ser. B 104: 437–464 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ioffe A.D. (2000). Metric regularity and subdifferential calculus. Russ. Math. Surv. 55: 501–558 zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Klatte D. and Kummer B. (2002). Constrained minima and Lipschitzian penalties in metric spaces. SIAM J. Optim. 13(2): 619–633 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Klatte D. and Kummer B. (2002). Nonsmooth Equations in Optimization—Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht zbMATHGoogle Scholar
  22. 22.
    Klatte D. and Kummer B. (2005). Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16: 96–119 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Klatte D. and Kummer B. (2006). Stability of inclusions: Characterizations via suitable Lipschitz functions and algorithms. Optimization 55: 627–660 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kummer B. (1998). Lipschitzian and pseudo-Lipschitzian inverse functions and applications to nonlinear programming. In: Fiacco, A.V. (eds) Mathematical Programming with Data Perturbations, pp 201–222. Marcel Dekker, New York Google Scholar
  25. 25.
    Kummer B. (1999). Metric regularity: characterizations, nonsmooth variations and successive approximation. Optimization 46: 247–281 zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kummer B. (2000). Inverse functions of pseudo regular mappings and regularity conditions. Math. Program. Ser. B 88: 313–339 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Li W. (1997). Abadie’s constraint qualification, metric regularity and error bounds for differentiable convex inequalities. SIAM J. Optim. 7: 966–978 zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lyusternik L. (1934). Conditional extrema of functions. Math. Sbornik 41: 390–401 zbMATHGoogle Scholar
  29. 29.
    Mordukhovich B.S. (1988). Approximation Methods in Problems of Optimization and Control (in Russian). Nauka, Moscow Google Scholar
  30. 30.
    Outrata J., Kočvara M. and Zowe J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer, Dordrecht zbMATHGoogle Scholar
  31. 31.
    Robinson S.M. (1976). Stability theorems for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13: 497–513 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Robinson S.M. (1979). Generalized equations and their solutions, Part I: Basic theory. Math. Program. Study 10: 128–141 zbMATHGoogle Scholar
  33. 33.
    Robinson S.M. (1980). Strongly regular generalized equations. Math. Oper. Res. 5: 43–62 zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Robinson S.M. (1982). Generalized equations and their solutions Part II: Applications to nonlinear programming. Math. Program. Study 19: 200–221 zbMATHGoogle Scholar
  35. 35.
    Robinson S.M. (2003). Variational conditions with smooth constraints: structure and analysis. Math. Program. 97: 245–265 zbMATHMathSciNetGoogle Scholar
  36. 36.
    Rockafellar R.T. and Wets R.J.-B. (1998). Variational Analysis. Springer, Berlin zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institut für Operations ResearchUniversität ZürichZürichSwitzerland
  2. 2.Institut für MathematikHumboldt–Universität zu BerlinBerlinGermany

Personalised recommendations