A local min-orthogonal method for finding multiple saddle points

The purpose of this paper is twofold. The first is to remove a possible ill-posedness related to a local minimax method developed in SIAMJ. Sci. Comput. 23 (2001) 840–865, SIAMJ. Sci. Comput. 24 (2002) 840–865 and the second is to provide a local characterization for nonminimax type saddle points. To do so, a local L-⊥ selection is defined and a necessary and sufficient condition for a saddle point is established, which leads to a min-orthogonal method. Those results exceed the scope of a minimax principle, the most popular approach in critical point theory. An example is given to illustrate the new theory. With this local characterization, the local minimax method in SIAM J. Sci. Comput. 23 (2001) 840–865, SIAM J. Sci. Comput. 24 (2002) 840–865 is generalized to a local min-orthogonal method for finding multiple saddle points. In a subsequent paper, this approach is applied to define a modified pseudo gradient (flow) of a functional for finding multiple saddle points in Banach spaces.  2003 Elsevier Inc. All rights reserved.