# Radius Theorems for Monotone Mappings

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## Abstract

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* : *X* → *X* 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.

### Keywords

Monotone mappings Maximal monotone Locally monotone Radius theorem Optimization problem Second-order sufficient optimality condition Newton method### Mathematics Subject Classification (2010)

47H05 49J53 90C31## Preview

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### References

- 1.Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)CrossRefMATHGoogle Scholar
- 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.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.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.Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrica
**1**, 211–218 (1936)CrossRefMATHGoogle Scholar - 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.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.Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res.
**27**, 170–191 (2002)MathSciNetCrossRefMATHGoogle Scholar - 9.Qi, Q.L., Sun, J.: A nonsmooth version of Newton’s method. Math. Programming A
**58**, 353–367 (1993)MathSciNetCrossRefMATHGoogle Scholar - 10.Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res.
**5**, 43–62 (1980)MathSciNetCrossRefMATHGoogle Scholar - 11.Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
- 12.Polioquin, R.A., Rockafellar, R.T.: Tilt stability of local minimum. SIAM J. Optim.
**8**(2), 287–299 (1998)MathSciNetCrossRefMATHGoogle Scholar - 13.Zolezzi, T.: A condition number theorem in convex programming. Math. Program.
**149**, 195–207 (2015)MathSciNetCrossRefMATHGoogle Scholar