Tobasco et al. [Physics Letters A, 382:382-386, 2018] recently suggested that trajectories of ODE systems which optimise the infinite-time average of a certain observable can be localised using sublevel sets of a function that arise when bounding such averages using socalled auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimisation is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localise the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations. We then combine these methods with a singleshooting Netwon-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimises the mean energy dissipation rate and maximises the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

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