Computation with Polynomial Equations and Inequalities arising in Combinatorial Optimization

The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear algebra or semidefinite programming relaxations of many kinds of feasibility or optimization questions. We are particularly interested in problems arising in combinatorial optimization.Comment: 28 pages, survey pape

The Ecosystem Approach to Fisheries: Issues, Terminology, Principles, Institutional Foundations, Implementation and Outlook

Ecosystems are complex and dynamic natural units that produce goods and services beyond those of benefit to fisheries. Because fisheries have a direct impact on the ecosystem, which is also impacted by other human activities, they need to be managed in an ecosystem context. The meaning of the terms 'ecosystem management', 'ecosystem based management', 'ecosystem approach to fisheries'(EAF), etc., are still not universally defined and progressively evolving. The justification of EAF is evident in the characteristics of an exploited ecosystem and the impacts resulting from fisheries and other activities. The rich set of international agreements of relevance to EAF contains a large number of principles and conceptual objectives. Both provide a fundamental guidance and a significant challenge for the implementation of EAF. The available international instruments also provide the institutional foundations for EAF. The FAO Code of Conduct for Responsible Fisheries is particularly important in this respect and contains provisions for practically all aspects of the approach. One major difficulty in defining EAF lies precisely in turning the available concepts and principles into operational objectives from which an EAF management plan would more easily be developed. The paper discusses these together with the types of action needed to achieve them. Experience in EAF implementation is still limited but some issues are already apparent, e.g. in added complexity, insufficient capacity, slow implementation, need for a pragmatic approach, etc. It is argued, in conclusion, that the future of EAF and fisheries depends on the way in which the two fundamental concepts of fisheries management and ecosystem management, and their respective stakeholders, will join efforts or collide

Positivity and optimization for semi-algebraic functions

We describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of Borel measurable functions, with particular emphasis to algebras generated by semi-algebraic functions. In which case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization. A collection of simple examples and numerical experiments complement the theoretical parts of the article.Comment: 20 page

Organisation spatiale des peuplements ichtyologiques des herbiers à Thalassia du Grand Cul-de-Sac Marin en Guadeloupe

Les herbiers à #Thalassia testudinum du Grand Cul-de-Sac Marin en Guadeloupe ont fait l'objet d'une étude de la variabilité spatiale de la faune ichtyologique et de quelques paramètres de l'environnement (densité, biomasse et longueur foliaire, agitation, température, oxygène dissous, pH et salinité). Les échantillons de poissons ont été prélevés à la senne de plage dans 11 stations réparties de la côte (mangrove) à la barrière frangeante des récifs coralliens, pendant 12 mois consécutifs (août 1987 à juillet 1988). Les stations non significativement différentes (test de Kruskal-Wallis) pour la température ou le pH, le sont en revanche, pour la densité, la biomasse et la longueur des feuilles de #Thalassia, l'oxygène dissous et la salinité, ainsi que pour la richesse spécifique, la régularité et la biomasse ichtyologiqu

Exploiting symmetries in SDP-relaxations for polynomial optimization

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.Comment: (v3) Minor revision. To appear in Math. of Operations Researc

Limitations of Algebraic Approaches to Graph Isomorphism Testing

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus

Systematic InSAR tropospheric phase delay corrections from global meteorological reanalysis data

6p.International audienceDespite remarkable successes achieved by Differential InSAR, estimations of low tectonic strain rates remain challenging in areas where deformation and topography are correlated, mainly because of the topography‐related atmospheric phase screen (APS). In areas of high relief, empirical removal of the stratified component of the APS may lead to biased estimations of tectonic deformation rates. Here we describe a method to correct interferograms from the effects of the spatial and temporal variations in tropospheric stratification by computing tropospheric delay maps coincident with SAR acquisitions using the ERA‐ Interim global meteorological model. The modeled phase delay is integrated along vertical profiles at the ERA‐I grid nodes and interpolated at the spatial sampling of the interferograms above the elevation of each image pixel. This approach is validated on unwrapped interferograms. We show that the removal of the atmospheric signal before phase unwrapping reduces the risk of unwrapping errors in areas of rough topography

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table