Derivative-Free Optimization Methods for Handling Fixed Costs in Optimal Groundwater Remediation Design [article]

Thomas Hemker, Kathleen Fowler, Thomas Hemker
Groundwater remediation design problems are routine in water resource management. The starting point for such a design problem is to formulate an objective function that represents a measure of the manager's goal. For example, in plume migration control, we need to determine the cost to design a well field to alter the direction of groundwater flow and thereby control the destination of a contaminant. Constraints must be specified to ensure that the plume is captured, the physical domain is
more » ... ected, and the wells operate under realistic conditions. Optimization algorithms must work in conjunction with groundwater flow and possibly contaminant transport simulators to determine the minimal cost well design subject to the constraints, but typically these numerical simulation codes have been developed for many years and have usually not been designed to meet the specific needs of optimization methods as, e.g., providing gradient information. Decision variables can be real-valued, in the case of pumping rates and well locations, or integer valued in the case of the number of wells in the design. In this work we focus on formulations that include a fixed installation cost as well as an operating cost, resulting in a simulation-based nonlinear mixed-integer optimization problem. The motivation is that our preliminary studies have shown that convergence to an unsatisfactory, local minimum with many wells operating at low pumping rates is common when the fixed cost is ignored. The challenge in the fixed cost formulation is the integer variable for the number of wells in the design. Removing a well from the design space leads to a large decrease in cost meaning optimizers must be equipped to either handle a mixed-integer or approximate mixed integer, black-box problem and discontinuities in the objective function. Moreover since evaluation of the objective function requires numerical results from a simulation, derivative information is unavailable. Gradient based optimization methods are not appropriate for these appli [...]
doi:10.4122/1.1000000575 fatcat:n57esn5gercl5awcvioxcvz5dy