Numerical optimal control of open channel networks From convex approximation to hidden invexity

Real-time management of water resources hinges upon four factors: (1) an understanding of the water balance, i.e., the inflows and extractions from a water system, (2) the amount of storage available in a water system, (3) an understanding of wave propagation through river and canal, i.e., open channel, systems, and (4) a way to accomodate multiple, potentially conflicting, uses.
This dissertation concerns itself with the challenge of real-time multi-objective optimal control of open channel systems, i.e., factors (3) and (4), while taking factors (1) and (2) into consideration.
The dynamics of water systems are nonlinear. Many real-time optimal control problems for water systems are therefore non-convex. From a mathematical point of view, this is problematic. In general, non-convex optimization problems admit an arbitrarily large number of local optima with different objective function values. Consequently, for such problems, the use of efficient local optimization methods is traditionally discouraged, and the use of expensive global optimization solvers, or of heuristic methods, is proposed instead.
In this dissertation, we address this non-convexity in two ways. The first approach is to accept a certain loss of model accuracy, and to approximate the non-convex problem with a convex problem.
The second approach is of a mathematically fundamental nature. This approach rests on a direct analysis of the quality of locally optimal solutions. The analysis is limited to a particular class of non-convex problems, for which it is proven that locally optimal solutions have strong nonlocal (global, or "nearly so") properties, a phenomenon referred to as "hidden invariant convexity".
The hidden invariant convexity results apply to a class of discrete-time optimal control problems. This class covers, but is not limited to, problems arising in water resources management. The result is particularly important because it shows that for this class of non-convex problems, heuristics or computationally expensive global solvers are not needed, and that local, deterministic methods — such as first and second-order methods — are sufficient in practice.