Mixing is an omnipresent process in a wide range of industrial applications, which supports scientific efforts to devise techniques for optimizing mixing processes under time and energy constraints. In this endeavour, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the nonlinear Navier–Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman penalization technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilize a Fourier-based approach for parameterizing and optimizing the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimization space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.