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For the first time, full-wave optimization exploiting adjoint Hessian matrices is applied to the design of microwave filters and transitions. The first- and second-order sensitivities of the scattering parameters are computed analytically using the adjoint network method (ANM). The mode-matching-based ANM is applied to the generalized scattering matrices of the different filter/transition components. Analytical gradient and Hessian matrices of differentiable objective functions are expressed in terms of the first- and second-order response adjoint sensitivities. Optimization techniques exploiting second-order information such as the Levenberg-Marquardt method are applied using the adjoint first- and second-order information. Significant acceleration is achieved using these techniques over gradient-based optimization techniques such as the Broyden-Fletcher-Goldfarb-Shanno method. The adjoint-based sensitivities are …