In this article, upper bounds on the worst-case H-2 performance index relative to structured, feedback perturbations are considered which are based on the minimisation of dual Lagrangean functionals over linearly-parametrised, finite-dimensional classes of dynamic multipliers. It is shown that the minimisation problems in question can be recast as optimisation problems with linear cost functional and Linear matrix inequality (LMI) constraints. An iterative scheme is suggested to generate linearly-parametrised classes of multipliers of increasing dynamic order so that progressively tighter upper bounds can be obtained, as illustrated by two simple numerical examples. Finally, with a view to synthesis procedures based on 'D-K iterations' relative to multipliers and controllers, it is shown that the minimisation of the upper-bounds corresponding to given multipliers with respect to linearly-parametrised classes of Youla parameters can also be cast as linear-cost/LMI-constraint problems.