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

Yao, Xudong

doi:10.1090/S0025-5718-2013-02669-5

A minimax method for finding saddle critical points of upper semi-differentiable locally Lipschitz continuous functional in Hilbert space and its convergence
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Math. Comp. 82 (2013), 2087-2136 Request permission

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

Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963. MR 1127041, DOI 10.1002/cpa.3160440808
K. C. Chang, The obstacle problem and partial differential equations with discontinuous nonlinearities, Comm. Pure Appl. Math. 33 (1980), no. 2, 117–146. MR 562547, DOI 10.1002/cpa.3160330203
Kung Ching Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102–129. MR 614246, DOI 10.1016/0022-247X(81)90095-0
Goong Chen, Jianxin Zhou, and Wei-Ming Ni, Algorithms and visualization for solutions of nonlinear elliptic equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), no. 7, 1565–1612. MR 1780923, DOI 10.1142/S0218127400001006
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Dimitrios Kandilakis, Nikolaos C. Kourogenis, and Nikolaos S. Papageorgiou, Two nontrivial critical points for nonsmooth functionals via local linking and applications, J. Global Optim. 34 (2006), no. 2, 219–244. MR 2210278, DOI 10.1007/s10898-005-3884-7
Nikolaos C. Kourogenis and Nikolaos S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance, J. Austral. Math. Soc. Ser. A 69 (2000), no. 2, 245–271. MR 1775181
Yongxin Li and Jianxin Zhou, A minimax method for finding multiple critical points and its applications to semilinear PDEs, SIAM J. Sci. Comput. 23 (2001), no. 3, 840–865. MR 1860967, DOI 10.1137/S1064827599365641
Yongxin Li and Jianxin Zhou, Convergence results of a local minimax method for finding multiple critical points, SIAM J. Sci. Comput. 24 (2002), no. 3, 865–885. MR 1950515, DOI 10.1137/S1064827500379732
Michael Struwe, Variational methods, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 34, Springer-Verlag, Berlin, 1996. Applications to nonlinear partial differential equations and Hamiltonian systems. MR 1411681, DOI 10.1007/978-3-662-03212-1
Xudong Yao and Jianxin 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, DOI 10.1007/s10107-005-0636-x
Xudong Yao and Jianxin Zhou, A minimax method for finding multiple critical points in Banach spaces and its application to quasi-linear elliptic PDE, SIAM J. Sci. Comput. 26 (2005), no. 5, 1796–1809. MR 2142597, DOI 10.1137/S1064827503430503
Xudong Yao and Jianxin Zhou, Unified convergence results on a minimax algorithm for finding multiple critical points in Banach spaces, SIAM J. Numer. Anal. 45 (2007), no. 3, 1330–1347. MR 2318815, DOI 10.1137/050627320