An efficient optimality test for the fuzzy c-means algorithm

The fuzzy c-means algorithm (FCM) is proved to converge to either local minimum or saddle point by Bezdek et al.(1987). However, it is problematical to judge the local minimum of a solution of the FCM in an easy way. In this paper, the Hessian matrix of one reduced objective function of the FCM is obtained and analyzed. Based on this study, a new optimality test of fixed points of the FCM is given, and its efficacy is verified by the examples in the paper. Moreover, a new stopping criterion for the FCM is also proposed.