Non-convex discrete-time optimal control problems in, e.g., water or power systems, typically involve a large number of variables related through nonlinear equality constraints. The ideal goal is to find a globally optimal solution, and numerical experience indicates that algorithms aiming for Karush–Kuhn–Tucker points often find solutions that are indistinguishable from global optima. In our paper, we provide a theoretical underpinning for this phenomenon, showing that on a broad class of problems the objective can be shown to be an invariant convex function (invex function) of the control decision variables when state variables are eliminated using implicit function theory. In this way, optimality guarantees can be obtained, the exact nature of which depends on the position of the solution within the feasible set. In a numerical example, we show how high-quality solutions are obtained with local search for a river control problem where invexity holds.

This study presents convex modeling of drainage pumps so that real-time control systems can be implemented to minimize their energy use. A convex model is built based on pump curves and then used in mixed-integer optimization to allow pumps to be turned on or off. It is implemented as an extension to the open source software package RTC-Tools. The formulation is such that the continuous relaxations of the mixed-integer problem are convex, hence branch-and-bound techniques may be used to find a global optimum. The formulation can be used for variable-speed and constant-speed pumps. There are several possible applications, such as optimization of polder systems, pumped-storage systems, or certain water distribution networks. Finally, an example of the drainage pump is presented to compare the method to current methods and show that energy can be saved by using the proposed method.