To account for endogeneity effects in, e.g., energy demand, modern bottom-up energy system models are often linked to a top-down model describing the macroeconomic system. Solving such linked models involves iteratively passing solutions from one model to the other and vice versa until convergence is reached, which can be computationally demanding. This paper proposes a coordination algorithm that speeds up convergence for the linkage of the two models in case the bottom-up model is a linear program and the top-down model is a mixed-complementarity problem. The coordination algorithm uses duality theory to select optimal bases from previous iterations to predict the solution of the bottom-up model. If the predicted solution is correct, which is shown to be equivalent to the predicted solution vector being non-negative, the bottom-up problem need not be solved in that iteration, resulting in a time gain. Numerical experiments on an energy system design problem illustrate that our coordination algorithm correctly predicts the bottom-up solution in most iterations, resulting in a significant reduction in overall computation time.