On global, local and stationary solutions in three-way data analysis

The issue of global and local solutions to optimization problems is of much interest in the context of three-way analysis, in particular when dealing with the PARAFAC and Tucker3 models or core transformations within the latt er. For clarity of statements, it is useful to consider the most simple yet reasonable situation, namely one-component PARAFAC decomposition or, close ly related, maximization of the leading squared core entry in Tucker3. In t he paper, necessary and sufficient conditions for global solutions are deri ved. Furthermore, it is shown that, in general, the usual cyclic co-ordinat e optimization scheme of three-way methods does not converge towards a loca l minimum (or maximum) even if the iterates yield global solutions in each co-ordinate direction. Finally, an example for a proper local minimum in on e-component PARAFAC is given. Copyright (C) 2000 John Wiley & Sons, Ltd.