Radius Theorems for Monotone Mappings



For a Hilbert space X and a mapping \(F: X\rightrightarrows X\) (potentially set-valued) that is maximal monotone locally around a pair \((\bar {x},\bar {y})\) in its graph, we obtain a radius theorem of the following kind: the infimum of the norm of a linear and bounded single-valued mapping B such that F + B is not locally monotone around \((\bar {x},\bar {y}+B\bar {x})\) equals the monotonicity modulus of F. Moreover, the infimum is not changed if taken with respect to B symmetric, negative semidefinite and of rank one, and also not changed if taken with respect to all functions f : XX that are Lipschitz continuous around \(\bar {x}\) and ∥B∥ is replaced by the Lipschitz modulus of f at \(\bar {x}\). As applications, a radius theorem is obtained for the strong second-order sufficient optimality condition of an optimization problem, which in turn yields a lower bound for the radius of quadratic convergence of the smooth and semismooth versions of the Newton method. Finally, a radius theorem is derived for mappings that are merely hypomonotone.


Monotone mappings Maximal monotone Locally monotone Radius theorem Optimization problem Second-order sufficient optimality condition Newton method 

Mathematics Subject Classification (2010)

47H05 49J53 90C31 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
  2. 2.
    Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc. 355, 493–517 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12, 79–109 (2004)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View From Variational Springer Monographs in Mathematics, 2nd edn. Springer, Dordrecht (2014)MATHGoogle Scholar
  5. 5.
    Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrica 1, 211–218 (1936)CrossRefMATHGoogle Scholar
  6. 6.
    Ioffe, A.D.: On stability estimates for the regularity property of maps. In: Topological Methods, Variational Methods and Their Applications, pp. 133–142. World Sci. Publ., River Edge (2003)Google Scholar
  7. 7.
    Mordukhovich, B.S., Nghia, T.T.A.: Local monotonicity and full stability for parametric variational systems. SIAM J. Optim. 26, 1032–1059 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 170–191 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Qi, Q.L., Sun, J.: A nonsmooth version of Newton’s method. Math. Programming A 58, 353–367 (1993)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  12. 12.
    Polioquin, R.A., Rockafellar, R.T.: Tilt stability of local minimum. SIAM J. Optim. 8(2), 287–299 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Zolezzi, T.: A condition number theorem in convex programming. Math. Program. 149, 195–207 (2015)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical ReviewsAnn ArborUSA
  2. 2.RMIT UniversityMelbourneAustralia
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations