A system is called equivariant if it is invariant with respect to a set of coordinate transformations associated to the elements of a multiplicative group. One established fact of the theory of equivariant systems is that various control problems can be solved by a generic controller if and only if they can be solved with a controller that satisfies the same invariance properties of the system. In this note, we show that this is true for all control tasks that can be obtained as a solution of an equivariant convex optimization problem and present some applications related to state and output feedback stabilization and decentralized control.