Version continue de l'algorithme d'Uzawa

In Carlier et al. (ESAIM Proceedings, CEMRACS 1999), an algorithm was proposed to approximate the projection of a function $\text{f∈H}{}^1{}_0\text{(Ω)}$ (where $\text{Ω}$ is a convex domain) onto the cone of convex functions. This algorithm is based on a dual expression of the constraint, which leads to a saddle-point problem which has no solution in general. We show here that the Uzawa algorithm for this saddle-point problem can be seen as the semi-discretization of an evolution equation

$\text{d}\text{λ}\text{d}\text{t}\text{+?Ψ(λ)?0,}$

where Ψ is a convex, l.s.c., proper function. In case the saddle-point problem has no solution, one has $\text{0∈}\text{R(?Ψ)}$ but ?Ψ^?1(0)=?. We establish that λ(t) is then divergent, and that a subsequence of the associated trajectory in the primal space converges weakly to the solution of the initial projection problem. To cite this article: B. Maury, C. R. Acad. Sci. Paris, Ser. I 337 (2003).