The four‐dimensional ensemble variational (4DEnVar) formulation is receiving increasing interest, especially in numerical weather prediction centres, which until now have mostly relied on the four‐dimensional variational (4D‐Var) formalism. It may indeed combine some of the best features of variational and ensemble methods. In this article, it is shown that the 4DEnVar formulation is linked with the 4D state formulation of variational assimilation, and that the 4DEnVar is relatively easy to precondition, in addition of being parallelizable. Practical implementations of the 4DEnVar are also investigated and two new preconditioned algorithms are proposed. The hybrid formulation of 4DEnVar, combining static and ensemble background‐error covariances, is discussed for the different possible algorithms. An application of the proposed implementations of 4DEnVar is shown with the Burgers model and compared to the use of 4D‐Var.

Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations even for non convex constraints. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. We first empirically compare this linearization approach with a previously proposed linearization approach of the literature on the continuous k−center problem. This methodology is then successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.

Because of the ever-increasing traffic and quality demands for both internet and television, satellite systems must aim at designs that use the satellite resources in the most efficient way possible. In the case of multibeam satellite systems, this is achieved by making optimal use of the plurality of beams in terms of frequency reuse, power allocation, and quality of the layout. That last point is the one addressed in this paper, the optimisation of the beam layout being a complex but crucial task for the resulting telecommunication system since it directly affects its performances and cost. In the case of broadband systems, the key data is the repartition of the traffic demand over the zone to be covered which is never rigorously uniform. Though, it is common for satellite system design tools to rely on this fairly unrealistic assumption to provide regular coverage which is therefore often suboptimal : inappropriate beamwidths, overprovisioned or unsatisfied user stations, unprofitable beams. Nonetheless, one strong advantage of the regular scheme is that it is known to be compatible with the single feed per beam antenna constraint of minimum angular distance for all the couples of beams coming from the same reflector. The aim of this paper is to present a randomized multi-start heuristic to build a non-uniform layout, driven by the different telecommunication mission criteria and by the aforementioned antenna constraint that is dealt with by a graph recoloring procedure via local search and simulated annealing.

International audienceIn a society where the demand for multimedia applications and data exchange is experiencing an unstoppable growth, multibeam systems have proven to be one of the most relevant solutions for satellite-based communication systems. Though already well represented among the geostationary satellites today, there are still several unresolved design optimization challenges for these complex systems that could lead to improved performances and to better system costs. The satellite platform, the repeater, and the antennas are examples of subsystems that should be designed jointly in order to reach an optimized technical solution that fulfills the service requirements. Traditionally, such complex tasks are addressed through a decomposition of the overall system design into a sequence of smaller decision problems. In this article, we propose to rely on operations research techniques to, on the one hand, take into account explicitly the in-terdependencies of these decomposed problems, and on the other hand, to handle the own constraints of each subsystem and their interactions. In this paper, the focus is laid on the optimization of the beam layouts of the multibeam satellites. Indeed, in addition to being a perfect example of the aforementioned importance of dealing with subsystem constraints, this problem appears early in the chain of design of a multibeam satellite system and is therefore critical for the quality of the telecommunication system: the weaknesses of a beam layout cannot be made up for later on in the system design. For this crucial optimization phase, the strength of the methodology we propose in this paper is to use mixed-integer linear programming to incorporate explicitly technological feasibility constraints of the subsystems involved, while preparing at best the subsequent design problems. Most importantly, our approach allows to overcome several resisting flaws of the already existing algorithms

This paper addresses an NP-hard design optimization problem in a multibeam satellite communication system. This problem consists in designing irregular beam layouts to satisfy non-uniform user traffic demands over the satellite service area, under antenna constraints, satellite payload constraints and a telecommunication mission criterion. Efficiently solving this problem is of crucial importance due to its impact on the system performance and cost. We compare three mixed-integer linear programming formulations. The first one, issued from previous work, is based on a linearization of both convex and non-convex Euclidean distance constraints. The two other aim at reducing the solution space size and at breaking symmetry inherent to the first formulation. For that purpose, we introduce a new process to interface k-means clustering with mixed-integer linear programming. We examine an exact and a heuristic approach for exploiting these principles that yield two new formulations. The heuristic approach outperforms the others based on our tests on a set of large-scale realistic problem instances, allowing to use mixed-integer linear programming in the industrial context.

Abstract:To comply with the continually growing demand for multimedia content and higher throughputs, the telecommunication industry has to keep improving the use of the bandwidth resources, leading to the well-known Frequency Assignment Problems (FAP). In this article, we present a new extension of these problems to the case of satellite systems that use a multibeam coverage. With the models we propose, we make sure that for each frequency plan produced there exists a corresponding satellite payload architecture that is cost-efficient and decently complex. Two approaches are presented and compared : a global constraint program that handles all the constraints simultaneously, and a decomposition method that involves both constraint programming and integer linear programming. For the latter approach, we show that the two identified subproblems can respectively be modeled as a multiprocessor scheduling problem and a path-covering problem, and this analogy is used to prove that they both belong to the category of NP-hard problems. We also show that, for the most common class of interference graphs in multibeam satellite systems, the maximal cliques can all be enumerated in polynomial time and their number is relatively low, therefore it is perfectly acceptable to rely on them in the scheduling model that we derived. Our experiments on realistic scenarios show that the decomposition method proposed can indeed provide a solution of the problem when the global CP model does not.

International audienceAs a result of the continually growing demand for multimedia content and higher throughputs in wireless communication systems, the telecommuni-cation industry has to keep improving the use of the bandwidth resources. This access to the radiofrequency spectrum is both limited and expensive, which has naturally lead to the definition of the generic class of combinatorial optimization problems known as " Frequency Assignment Problems " (FAP). In this article, we present a new extension of these problems to the case of satellite systems that use a multibeam coverage. With the models we propose, we make sure that for each frequency plan produced there exists a corresponding satellite payload architecture that is cost-efficient and decently complex. Two approaches are presented and compared : a global constraint program that handles all the constraints simultaneously , and a decomposition method that involves both constraint programming and integer linear programming. For the latter approach where two subprob-lems are studied, we show that one of them can be modeled as a multiprocessor scheduling problem while the other can either be seen as a path-covering problem or a multidimensionnal bin-packing problem depending on the assumptions made. These analogies are used to prove that both the subproblems addressed in the decomposition method belong to the category of NP-hard problems. We also show that, for the most common class of interference graphs in multibeam satellite systems, the maximal cliques can all be enumerated in polynomial time and their number is relatively low, therefore it is perfectly acceptable to rely on them in the scheduling model that we derived. Our experiments on realistic scenarios show that the decomposition method proposed can indeed provide a solution of the problem when the global CP model does not