Multidimensional Scaling with City-Block Distances

Multidimensional scaling is a technique for exploratory analysis of multidimensional data. The essential part of the technique is minimization of a function with unfavorable properties like multimodality, invariants and non-differentiability. Recently various two-level optimization algorithms for multidimensional scaling with city-block distances have been proposed exploiting piecewise quadratic structure of the least squares objective function with such distances. A problem of combinatorial optimization is tackled at the upper level, and convex quadratic programming problems are tackled at the lower level. In this paper we discuss a new reformulation of the problem where lower level quadratic programming problems seem more suited for two-level optimization.