The mixing of binary fluids by stirring is a ubiquitous process in a wide range of industrial applications. The food processing, pharmaceutical and consumer product industries are dominated by mixing processes, and enhancing mixing efficiency for these applications, even by a very modest amount, would translate into considerable savings and benefits. We develop and use an adjoint-based nonlinear optimization scheme to minimize the mix-norm of a passive scalar. The mixing is accomplished by embedded cylindrical stirrers, either fixed rotating or moving on concentric circular paths inside a circular container. We consider three separate optimising regimes, where (i) the eccentricity and rotational velocity, (ii) the cross sectional shape and (iii) the velocities of the stirrers are optimized to reach improved mixedness over a finite time horizon. In all cases, the enhanced stirring protocol is shown to consist of a complicated interplay of vortical structures which have been created and exploited by the stirrers’ action. In all scenarios, substantial mixing enhancement could be achieved. Further extensions of the mathematical formalism and the physical setup will be discussed, and remaining challenges of this research effort will be addressed.