Fuzzy c-means: Optimality of solutions and effective termination of the algorithm

In this paper, the solutions produced by the fuzzy c-means algorithm for a general class of problems are examined and a method to test for the local optimality of such solutions is established. An equivalent mathematical program is defined for the c-means problem utilizing a generalized norm, then the properties of the resulting optimization problem are investigated. It is shown that the gradient of the resulting objective function at the solution produced by the c-means algorithm in this case takes a special structure which can be used in terminating the algorithm. Moreover, the local optimality of the solution obtained is checked utilizing the Hessian of the criterion function. The solution is a local minimum point if the Hessian matrix at this point is positive semidefinite. Simple rules are proposed to help in checking the definiteness of the matrix.