A minimax linear quadratic (LQ) output-feedback controller is introduced which minimises the maximal value of the performance index over all initial states belonging to some set separated out a priori. If the set is an ellipsoid or a polygon, such controllers are synthesised in terms of linear matrix inequalities (LMIs). In particular case when a size of this set tends to zero tightening to a point, the minimax LQ controller approaches the optimal LQ output-feedback controller for the given initial state, while in another extreme case when this size tends to infinity, we have the worst-case LQ output-feedback controller. Numerical results for an inverted pendulum are presented.