On the Existence of Nonglobal Minimizers of the Stress Criterion for Metric Multidimensional Scaling

tive deenite and X is a local minimizer of. To the best of our knowledge, this is the rst formal demonstration of a nonglobal minimizer of the stress criterion. 4 Discussion We now know that nonglobal minimizers of the stress criterion can exist. However, because of the tendency of SMACOF-I to terminate prematurely, the prevalence of nonglobal minimizers may have been overestimated in previous reports. More accurate assessments will require careful numerical experimentation. The prevalence of nonglobal minimizers is an interesting issue, but an issue of greater practical importance is the prevalence of failure to converge to a global minimizer. Although these issues are obviously related, our numerical experience suggests that they are not identical. We performed informal experiments with SMACOF-1 that suggested that the nonglobal minimizer identiied in Example 2 has a fairly large basin of attraction. Anthony Kearsley performed analogous experiments with the globalized Newton's method described in Section 2.2 and consistently converged to a global minimizer, even from initial conngurations quite close to the nonglobal min-imizer. We do not completely understand this paradox , but perhaps the basins of nonglobal minimizers of the stress criterion are so shallow that Newton's method tends to step over them. This remarkable possibility certainly deserves investigation.