Unified Convergence Results on a Minimax Algorithm for Finding Multiple Critical Points in Banach Spaces
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A minimax method for finding multiple critical points in Banach spaces is successfully developed in [X. Yao and J. Zhou, SIAM J. Sci. Comput., 26 (2005), pp. 1796-1809] by using a projected pseudogradient as a search direction. Since several different techniques can be used to compute a projected pseudogradient, the uniform stepsize and the continuity of a search direction, two key properties for the convergence results in [Y. Li and J. Zhou, SIAM J. Sci. Comput., 24 (2002), pp. 865-885], get lost. In this paper, instead of proving convergence results of the algorithm for each technique, unified convergence results are obtained with a weaker stepsize assumption. An abstract existence-convergence result is also established. It is independent of the algorithm and explains why function values always converge faster than their gradients do. The weaker stepsize assumption is then verified for several different cases. As an illustration to the new results, the Banach space W 01,p () is considered and the conditions posed in the new results are verified for a quasi-linear elliptic PDE. 2007 Society for Industrial and Applied Mathematics.
