Journal of Optimization Theory and Applications

, Volume 158, Issue 1, pp 216–233

# Convergence Analysis of the Gauss–Newton-Type Method for Lipschitz-Like Mappings

Article

## Abstract

We introduce in the present paper a Gauss–Newton-type method for solving generalized equations defined by sums of differentiable mappings and set-valued mappings in Banach spaces. Semi-local convergence and local convergence of the Gauss–Newton-type method are analyzed.

## Keywords

Set-valued mappings Lipschitz-like mappings Generalized equations Gauss–Newton-type method Semi-local convergence

## Notes

### Acknowledgements

The authors thank the referees and the associate editor for their valuable comments and constructive suggestions which improved the presentation of this manuscript. Research work of the first author is fully supported by Chinese Scholarship Council, and research work of the third author is partially supported by National Natural Science Foundation (grant 11171300) and Zhejiang Provincial Natural Science Foundation (grant Y6110006) of China.

## References

1. 1.
Robinson, S.M.: Generalized equations and their solutions, part I: basic theory. Math. Program. Stud. 10, 128–141 (1979)
2. 2.
Robinson, S.M.: Generalized equations and their solutions, part II: applications to nonlinear programming. Math. Program. Stud. 19, 200–221 (1982)
3. 3.
Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)
4. 4.
Dontchev, A.L.: Local convergence of the Newton method for generalized equation. C. R. Acad. Sci. Paris, Ser. I 322, 327–331 (1996)
5. 5.
Dedieu, J.P., Kim, M.H.: Newton’s method for analytic systems of equations with constant rank derivatives. J. Complex. 18, 187–209 (2002)
6. 6.
Dedieu, J.P., Shub, M.: Newton’s method for overdetermined systems of equations. Math. Comput. 69, 1099–1115 (2000)
7. 7.
Li, C., Zhang, W.H., Jin, X.Q.: Convergence and uniqueness properties of Gauss–Newton’s method. Comput. Math. Appl. 47, 1057–1067 (2004)
8. 8.
He, J.S., Wang, J.H., Li, C.: Newton’s method for underdetemined systems of equations under the modified γ-condition. Numer. Funct. Anal. Optim. 28, 663–679 (2007)
9. 9.
Xu, X.B., Li, C.: Convergence of newton’s method for systems of equations with constant rank derivatives. J. Comput. Math. 25, 705–718 (2007)
10. 10.
Xu, X.B., Li, C.: Convergence criterion of Newton’s method for singular systems with constant rank derivatives. J. Math. Anal. Appl. 345, 689–701 (2008)
11. 11.
Robinson, S.M.: Extension of Newton’s method to nonlinear functions with values in a cone. Numer. Math. 19, 341–347 (1972)
12. 12.
Li, C., Ng, K.F.: Majorizing functions and convergence of the Gauss–Newton method for convex composite optimization. SIAM J. Optim. 18, 613–642 (2007)
13. 13.
Aubin, J.P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)
14. 14.
Jean-Alexis, C., Piétrus, A.: On the convergence of some methods for variational inclusions. Rev. R. Acad. Cien. Serie A. Mat. 102, 355–361 (2008)
15. 15.
Argyros, I.K., Hilout, S.: Local convergence of Newton-like methods for generalized equations. Appl. Math. Comput. 197, 507–514 (2008)
16. 16.
Dontchev, A.L.: The Graves theorem revisited. J. Convex Anal. 3, 45–53 (1996)
17. 17.
Haliout, S., Piétrus, A.: A semilocal convergence of a secant-type method for solving generalized equations. Positivity 10, 693–700 (2006)
18. 18.
Pietrus, A.: Does Newton’s method for set-valued maps converges uniformly in mild differentiability context? Rev. Colomb. Mat. 34, 49–56 (2000)
19. 19.
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I: Basic Theory. Springer, Berlin (2006) Google Scholar
20. 20.
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990)
21. 21.
Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design. SIAM Proc. Appl. Math., vol. 58, pp. 32–46 (1992) Google Scholar
22. 22.
Penot, J.P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13, 629–643 (1989)
23. 23.
Dontchev, A.L., Hager, W.W.: An inverse mapping theorem for set-valued maps. Proc. Am. Math. Soc. 121, 481–489 (1994)
24. 24.
Burke, J.V., Ferris, M.C.: A Gauss–Newton method for convex composite optimization. Math. Program. 71, 179–194 (1995)
25. 25.
Li, C., Wang, X.H.: On convergence of the Gauss–Newton method for convex composite optimization. Math. Program., Ser. A 91, 349–356 (2002)
26. 26.
Dontchev, A.L.: Uniform convergence of the Newton method for Aubin continuous maps. Serdica Math. J. 22, 385–398 (1996)