Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

A minimax method for finding saddle critical points of upper semi-differentiable locally Lipschitz continuous functional in Hilbert space and its convergence


Author: Xudong Yao
Journal: Math. Comp. 82 (2013), 2087-2136
MSC (2010): Primary 65K10, 65K15, 65N12; Secondary 49M37
DOI: https://doi.org/10.1090/S0025-5718-2013-02669-5
Published electronically: March 28, 2013
MathSciNet review: 3073193
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A minimax characterization for finding nonsmooth saddle critical points, i.e., saddle critical points of locally Lipschitz continuous functional, in Banach space is presented in [X. Yao and J. Zhou, A local minimax characterization for computing multiple nonsmooth saddle critical points, Math. Program., 104 (2005), no. 2-3, Ser. B, 749-760]. By this characterization, a descent-max method is devised. But, there is no numerical experiment and convergence result for the method. In this paper, to a class of locally Lipschitz continuous functionals, a minimax method for computing nonsmooth saddle critical points in Hilbert space will be designed. Numerical experiments will be carried out and convergence results will be established.


References [Enhancements On Off] (What's this?)

  • 1. ROBERT ALEXANDER ADAMS AND JOHN J. F. FOURNIER, Sobolev Spaces, Academic Press, New York, 2003. MR 2424078 (2009e:46025)
  • 2. A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. MR 0370183 (51:6412)
  • 3. H. Brezis and L. Nirenberg, Remarks on finding critical points, Commun. Pure Appl. Math., 44 (1991), 939-963. MR 1127041 (92i:58032)
  • 4. K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math., 33 (1980), 117-146. MR 562547 (81f:35032)
  • 5. K. C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. MR 614246 (82h:35025)
  • 6. G. Chen, W. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations Part I: Dirichlet problems, Int. J. Bifurcation $ \&$ Chaos, 7 (2000), 1565-1612. MR 1780923
  • 7. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. MR 709590 (85m:49002)
  • 8. Dimitrios Kandilakis, Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou, Two nontrival critical points for nonsmooth functionals via local linking and applications, J. Global. Optim., 34 (2006), 219-244. MR 2210278 (2007b:35081)
  • 9. Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Aust. Math. Soc., A 69 (2000), 245-271. MR 1775181 (2001m:35078)
  • 10. Y. Li and J. Zhou, A minimax method for finding multiple critical points and its applications to nonlinear PDEs, SIAM J. Sci. Comput., 23 (2001), 840-865. MR 1860967 (2002h:49012)
  • 11. Y. Li and J. Zhou, Convergence results of a local minimax method for finding multiple critical points, SIAM J. Sci. Comput., 24 (2002), 865-885. MR 1950515 (2004a:58013)
  • 12. M. Struwe, Variational Methods, Springer-Verlag, New York, 1996. MR 1411681 (98f:49002)
  • 13. X. Yao and J. Zhou, A local minimax characterization for computing multiple nonsmooth saddle critical points, Math. Program., 104 (2005), no. 2-3, Ser. B, 749-760. MR 2179258 (2006k:90166)
  • 14. X. Yao and J. Zhou, A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE, SIAM J. Sci. Comp., 26 (2005), 1796-1809. MR 2142597 (2006a:58014)
  • 15. X. Yao and J. Zhou, Unified convergence results on a minimax algorithm for finding multiple critical points in Banach spaces, SIAM J. Num. Anal., 45 (2007), 1330-1347. MR 2318815 (2009c:65139)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65K10, 65K15, 65N12, 49M37

Retrieve articles in all journals with MSC (2010): 65K10, 65K15, 65N12, 49M37


Additional Information

Xudong Yao
Affiliation: Department of Mathematic, Shanghai Normal University, Shanghai, China 200234
Email: xdyao@shnu.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-2013-02669-5
Keywords: Locally Lipschitz continuous functional, nonsmooth saddle critical point, minimax method, convergence
Received by editor(s): July 23, 2010
Received by editor(s) in revised form: October 23, 2011
Published electronically: March 28, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society