Solving constrained Procrustes problems: a conic optimization approach

Procrustes problems are matrix approximation problems searching for a~transformation of the given dataset to fit another dataset. They find applications in numerous areas, such as factor and multivariate analysis, computer vision, multidimensional scaling or finance. The known methods for solving Procrustes problems have been designed to handle specific sub-classes, where the set of feasible solutions has a special structure (e.g. a Stiefel manifold), and the objective function is defined using a specific matrix norm (typically the Frobenius norm). We show that a wide class of Procrustes problems can be formulated and solved as a (rank-constrained) semi-definite program. This includes balanced and unbalanced (weighted) Procrustes problems, possibly to a partially specified target, but also oblique, projection or two-sided Procrustes problems. The proposed approach can handle additional linear, quadratic, or semi-definite constraints and the objective function defined using the Frobenius norm but also standard operator norms. The results are demonstrated on a set of numerical experiments and also on real applications

A unified approach to radial, hyperbolic, and directional distance models in Data Envelopment Analysis

The paper analyzes properties of a large class of "path-based" Data Envelopment Analysis models through a unifying general scheme, which includes as standard the well-known oriented radial models, the hyperbolic distance measure model, and the directional distance measure models. The scheme also accommodates variants of standard models over negative data. Path-based models are analyzed from the point of view of nine desired properties that a well-designed model should satisfy. The paper develops mathematical tools that allow systematic investigation of these properties in the general scheme including, but not limited to, the standard path-based models. Among other results, the analysis confirms the generally accepted view that path-based models need not generate Pareto--Koopmans efficient projections, one-to-one identification, or strict monotonicity