Abstract: During the last decade, Kriging-based sequential algorithms like EGO and its variants have become reference optimization methods in computer experiments. Such algorithms rely on the iterative maximization of a sampling criterion, the expected improvement (EI), which takes advantage of Kriging conditional distributions to make an explicit trade-off between promizing and uncertain search space points. We have recently worked on a multipoints EI criterion meant to simultaneously choose several points, which is useful for instance in synchronous parallel computation. The research results that we wish to present in this paper concern sequential procedures with a fixed number of iterations. We show that maximizing the 1-point criterion at each iteration (EI algorithm) is suboptimal. In essence, the latter amounts to considering the current iteration as the last one. This work formulates the problem of optimal strategy for finite horizon sequential optimization, provides the solution to this problem in terms of multipoints EI, and illustrates the suboptimality of the usual EI algorithm on the basis of a first counter-example.