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When formulating a stochastic optimization problem, one has to know the probability distributions of all involved random elements. However, as the typical source of information about these distributions are some observed past data, the estimated distributional model is subject to estimation error. This uncertainty about the correct model is called the "ambiguity problem". We propose a minimax approach where in the first step an appropriate confidence region is determined by statistical methods and then a minimax approach is adopted in the subsequent optimization step. Our main example will be from portfolio optimization under a convex risk constraint. Here the minimax problem can be solved by sequential linear programming (SLP). It turns out that the obtained solutions are robust w.r.t. to model error and the cost of this robustness in terms of the loss in performance compared to the non-ambiguous problem is rather small.