Semidefinite Programming Approach to Russell Measure Model

Throughout its evolution, data envelopment analysis (DEA) has mostly relied on linear programming, particularly because of simple primal-dual relations and the existence of standard software for solving linear programs. Although also nonlinear models, such as Russell measure or hyperbolic measure models, have been introduced, their use in applications has been limited mainly because of their computational inconvenience. The common feature of these nonlinear models is that some unknown variables appear in the form of reciprocal values. In this paper, we introduce a novel method for dealing with this type of nonlinearity in DEA. We show how to reformulate the nonlinear model as a semidefinite programming (SDP) problem and describe how to derive the corresponding dual counterpart of the model. Two benefits of our approach are: (1) the SDP reformulated model can be solved efficiently using some standard SDP solvers and, (2) the derived dual program is comparable with the dual counterparts of linear DEA models and allows the common economic interpretations of dual variables as shadow prices. Our approach is applied to the Russell measure model for that also an overview of attempts to overcome the nonlinearities is presented.