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Recently, we proposed a novel technique for design sensitivity analysis of high-frequency structures with respect to localized perturbations in conductive parameters. Here, we generalize the technique to include shape and material variations and utilize the response sensitivities in gradient-based optimization. Our technique belongs to the class of adjoint-variable methods. Thus, it computes the response and its gradient with only two electromagnetic (EM) simulations-of the original and the adjoint problems-regardless of the number of design parameters. For the first time, adjoint sensitivities with respect to conductive, dielectric-magnetic material and shape perturbations are computed via EM solvers on structured grids. Our approximate sensitivity analysis does not require analytical derivatives of the system matrix. This makes the technique versatile and easy to implement. The technique defaults to exact sensitivities with analytical system matrix derivatives when global design parameters are being perturbed. We discuss the accuracy of the approximate sensitivities, as well as the practicality of the exact sensitivities in specific design problems. We also discuss implementations in gradient-based optimization and illustrate them through simulation and design with the frequency domain transmission line method (FD-TLM).