Within this work, we present a warm-started algorithm for Model Predictive Control (MPC) of switched nonlinear systems under combinatorial constraints based on Combinatorial Integral Approximation (CIA). To facilitate high-speed solutions, we introduce a preprocessing step for complexity reduction of CIA problems, and include this approach within a new toolbox for solution of CIA problems with special focus on MPC. The proposed algorithm is implemented and utilized within an MPC simulation study for a solar thermal climate system with nonlinear system behavior and uncertain operation conditions. The results are analyzed in terms of solution quality, constraint satisfaction and runtime of the solution steps, showing the applicability of the proposed algorithm and implementations. (C) 2019 The Author(s). Published by Elsevier Ltd.