In this paper, we present a generalized mixed-sensitivity multivariable framework for linear time invariant (LTI) plants that can handle a broad class of closed loop (e.g. hamilt∞, hamilt2, frequency- and time domain) objectives while being able to directly and systematically address the problem of trading off properties at distinct loop breaking points. This is done by exploiting the Youla-Jabr-Bongiorno-Kucera-Zames (YJBKZ) parameterization, the resulting convexification, and efficient convex solvers that can be applied to smooth as well as non-differentiable problems. Our approach is shown to be particularly useful for ill-conditioned plant having large relative gain array entries - plants that have received considerable attention in the literature without yielding a direct systematic design methodology. Moreover, we also show how our approach can be applied to multivariable infinite-dimensional plants. We specifically show that by suitably approximating the infinite-dimensional plant with a finite-dimensional approximant, a near-optimal finite-dimensional controller can be designed for the infinite-dimensional plant. Illustrative examples are provided.