This paper presents a novel gradient based iterative approach to concurrently optimize plant and controller parameters. The co-design approach is linear matrix inequality (LMI) based and relies on estimates of the differential properties of the objective function with respect to the design variables which are obtained by using the dual solution to the control design problem. Contrary to most existing techniques which assume an affine dependency of the objective function on the optimization problem, the presented approach allows for nonlinear dependency, and hence can handle a broader class of co-design problems. Moreover, unlike other gradient based techniques, the method doesn't need to compute controller parameters at each iteration. The developed method is validated using numerical examples of a gantry machine and an electric motor.