Extensions of Classical Multidimensional Scaling: Variable Alternation and Nonconvex Duality∗

Classical multidimensional scaling constructs a configuration of points that minimizes a certain measure of discrepancy between the configuration’s interpoint distance matrix and a fixed dissimilarity matrix. Recent extensions have replaced the fixed dissimilarity matrix with a closed and convex set of dissimilarity matrices. These extensions lead to optimization problems with two sets of decision variables, distance variables and dissimilarity variables. A popular optimization strategy that attempts to exploit the structure of such problems is variable alternation, in which one alternates holding one set of variables fixed and optimizing with respect to the other. We study the theory of variable alternation for extensions of classical multidimensional scaling and connect this approach with an approach based on nonconvex duality. As part of this comparison, we identify the objective function as the difference of convex functions (a d.c. function), which allows us to compute its nonconvex dual. This, in turn, leads to a new Lagrangian formulation of the problem.