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We consider a class of large scale robust optimization problems. While the robust optimization literature often relies on structural assumptions to reformulate the problem in a tractable form using duality, this method is not always applicable and can result in problems which are very large. We propose an alternative way of solving such problems by applying a constrained bundle method. The originality of the method lies in the fact that the minimization steps in the bundle method are solved approximately using the alternating direction method of multipliers. Numerical results from a power grid regulation problem are presented and support the relevance of the approach.