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Optimization problems with Hammerstein\u2013Wiener models embedded are nonconvex, and thus local optimization methods may obtain suboptimal solutions. In this work, we develop a deterministic global optimization strategy that exploits the specific structure of Hammerstein\u2013Wiener models to extend existing theory on global optimization of systems with linear dynamics. At first, we discuss alternative formulations of the dynamic optimization problem with Hammerstein\u2013Wiener models embedded, demonstrating that careful selection of the optimization variables of the problem can offer significant numerical advantages to the solution approach. Then, we develop convex relaxations for the proposed optimization problem and discuss implementation aspects to obtain the global solution focusing on a control parametrization technique. Finally, we apply our optimization strategy to case studies comprising both offline and online dynamic optimization problems. The results confirm an improved computational performance of the proposed solution approach over alternative options not exploiting the linear dynamics for all considered examples. 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