For a linear time-invariant system with several disturbance inputs and controlled outputs, we show how to minimize the nominal H-2-norm performance in one channel while keeping bounds on the H-2-norm or H-infinity-norm performance (implying robust stability) in the other channels, This multiobjective H-2/H-infinity-problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems. We show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller, If existing, we reveal how the novel trick of optimizing the trace norm of the Youla parameter over certain convex constraints allows to design a nearly optimal controller whose Youla parameter is of the same order as the optimal one.