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.