Minimum L1-distance projection onto the boundary of a convex set: Simple characterization

We show that the minimum distance projection in the L1-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec.Comment: 5 page

Technology Adoption in French Agriculture and the Role of Financial Constraints

Successive CAP reforms have increased the exposure of European agriculture to market forces. As a result, farmers have become preoccupied with their competitiveness and have progressively adopted best practices. However, these long-run technological adjustments could be slowed down by eventual shortrun financial constraints. This contribution measures the role of these financial constraints on the catching-up component of total factor productivity for a panel of French farmers in Nord-Pas-de-Calais region during 1994-2001. For TFP estimates based on non-parametric distance functions, the second stage econometric results indicate that the technological adaptation is significantly conditioned by financial constraints.TFP catching-up, distance function, financial constraints, Agricultural Finance, Farm Management, Research and Development/Tech Change/Emerging Technologies,

Tropical Fourier-Motzkin elimination, with an application to real-time verification

We introduce a generalization of tropical polyhedra able to express both strict and non-strict inequalities. Such inequalities are handled by means of a semiring of germs (encoding infinitesimal perturbations). We develop a tropical analogue of Fourier-Motzkin elimination from which we derive geometrical properties of these polyhedra. In particular, we show that they coincide with the tropically convex union of (non-necessarily closed) cells that are convex both classically and tropically. We also prove that the redundant inequalities produced when performing successive elimination steps can be dynamically deleted by reduction to mean payoff game problems. As a complement, we provide a coarser (polynomial time) deletion procedure which is enough to arrive at a simply exponential bound for the total execution time. These algorithms are illustrated by an application to real-time systems (reachability analysis of timed automata).Comment: 29 pages, 8 figure

Minimal half-spaces and external representation of tropical polyhedra

We give a characterization of the minimal tropical half-spaces containing a given tropical polyhedron, from which we derive a counter example showing that the number of such minimal half-spaces can be infinite, contradicting some statements which appeared in the tropical literature, and disproving a conjecture of F. Block and J. Yu. We also establish an analogue of the Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently represented internally (in terms of extreme points and rays) or externally (in terms of half-spaces containing it). A canonical external representation of a polyhedron turns out to be provided by the extreme elements of its tropical polar. We characterize these extreme elements, showing in particular that they are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures improved, references update

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Eco-efficiency measurement and material balance principle:an application in power plants Malmquist Luenberger Index

Incorporating Material Balance Principle (MBP) in industrial and agricultural performance measurement systems with pollutant factors has been on the rise in recent years. Many conventional methods of performance measurement have proven incompatible with the material flow conditions. This study will address the issue of eco-efficiency measurement adjusted for pollution, taking into account materials flow conditions and the MBP requirements, in order to provide ‘real’ measures of performance that can serve as guides when making policies. We develop a new approach by integrating slacks-based measure to enhance the Malmquist Luenberger Index by a material balance condition that reflects the conservation of matter. This model is compared with a similar model, which incorporates MBP using the trade-off approach to measure productivity and eco-efficiency trends of power plants. Results reveal similar findings for both models substantiating robustness and applicability of the proposed model in this paper

A classification of DEA models when the internal structure of the Decision Making Units is considered

We classify the contributions of DEA literature assessing Decision Making Units (DMUs) whose internal structure is known. Starting from an elementary framework, we define the main research areas as shared flow, multilevel and network models, depending on the assumptions they are subject to. For each model category, the principal mathematical formulations are introduced along with their main variants, extensions and applications. We also discuss the results of aggregating efficiency measures and of considering DMUs as submitted to a central authority that imposes constraints or targets on them. A common feature among the several models is that the efficiency evaluation of the DMU depends on the efficiency values of its subunits thereby increasing the discrimination power of DEA methodology with respect to the black box approach