In this paper an optimal control problem is considered in which the cost functional is the sum of the H-2-norms of several closed-loop transfer functions and the stabilizing controllers are constrained to satisfy a prescribed upper limit on the H-infinity-norm of another transfer function. An iterative procedure is described which aims at generating approximate solutions to this problem through the solutions of a sequence of H-2-cost/H-2-constraint optimization problems in which the H-2-constraint is iteratively modified so as to increase the resulting optimal value of the original cost functional. Conditions are presented under which such an H-2/H-2 problem is shown to have a unique (rational) solution; it is also shown that the solution can be computed by means of line search and spectral factorization. It turns out that the sequence of optimal cost values of the H-2/H-2 problems is monotonically increasing; and whenever it approaches the optimal value for the original H-2/H-infinity problem, the corresponding controllers tend ’approximately’ to satisfy the H-infinity constraint. The approximate solutions obtained in this process are illustrated by three numerical examples.