Aerothermal optimization is a powerful technique for automating the design of turbine internal cooling passages, where both pressure loss and heat transfer are considered. Existing optimization studies commonly adopt gradient-free algorithms, which can handle only a few design variables. However, to enhance heat transfer, internal cooling designs use complex geometries consisting of ribbed serpentine channels, which need to be parameterized by using a large number of design variables. To address this need, we develop herein an approach for aerothermal optimization that uses a gradient-based optimizer in conjunction with a discrete adjoint method to efficiently compute the required gradients with respect to numerous design variables. We apply this approach to the design of a ribbed U-bend channel, which is representative of a section of turbine internal cooling passages. The objective function combines heat transfer and pressure loss as a weighted sum. We find the Pareto front for these two objectives by running five optimizations with different weights. We consider both a rib-free and a ribbed U-bend configuration. For the rib-free configuration, we use 113 design variables to parameterize the U-bend shape. We compare our optimization results with those from gradient-free methods and demonstrate that the proposed method leads to lower pressure loss while enhancing heat transfer. For the ribbed configuration, we use 146 design variables and allow the ribs to change their arrangement independently (shape, height, pitch, and angle). Each rib adopts a requisite arrangement to balance heat transfer and aerodynamics, depending on the local flow conditions. Optimizing the U-bend shape is shown to be more effective for improving overall heat transfer than optimizing the rib arrangement. However, optimizing ribs is more effective for improving local heat transfer. The results demonstrate that the proposed optimization framework has the potential to handle general turbine heat transfer designs, not only for internal cooling but also for other design problems, such as film cooling and jet impingement cooling. (C) 2019 Elsevier Ltd. All rights reserved.