A class of matrix function optimization problems is considered in which the cost functional is the H-2-norm of an affine function of the decision variables and there are several constraints involving H-2 and H-infinity norms of other affine functions. Such problems arise naturally in feedback control theory as closed-loop transfer functions are affine functions of the so-called Youla parameter and control objectives can often be expressed as prescribed upper bounds on the H-2 and H-infinity norms of these functions. A procedure for obtaining approximate solutions to such problems is described which involves solving a sequence of H-2 problems with a single weighted H-2 constraint. This procedure hinges upon a weighting updating scheme in which, roughly speaking, a multiplication operator is applied to an additively corrected version of the weighting function at each step, with both the operator and the additive correction being defined on the basis of the solution of the current, auxiliary quadratic problem. It is shown that the sequence of optimal cost values for the auxiliary H-2/H-2 problems is monotonically non-decreasing and bounded from above (hence, it is convergent); and that it fails to increase in a given step if and only if the corresponding solution is optimal for the original multi-objective H-2/H-infinity problem-thus, heuristically speaking, as the auxiliary cost increments tend to zero, the auxiliary solutions become approximately optimal for the original problem. Indeed, conditions are established under which the procedure in question generates approximate solutions to the original H-2 problem with several H-2 and H-infinity constraints. This is based on a lower bound on the one-step increments of the sequence of auxiliary optimal cost values which is a function of a real-valued parameter associated with the additive-correction scheme. To improve convergence, this parameter can be chosen in such a way as to maximize the lower bound involved.