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

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 protected, 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 applications, hence methods that rely only on function values are more appealing. We compare three methods for handling the installation cost on a hydraulic capture benchmarking problem proposed in the literature. All the approaches described below do not use the gradient of the objective function, only function values for minimization. In one approach, we use penalty coefficients proposed in the literature for the installation term to transform the discontinuous problem into a continuous one. In another approach, we bypass including the number of wells as a decision variable by defining an inactive-well threshold. In the course of the optimization, if a well rate falls in this threshold, the well is removed from the design space, leading to large discontinuities in the objective function. For the two above formulations, we use the implicit filtering algorithm, a method which uses a sequence of finite difference gradients, for minimization. In the third approach, we use sequential stochastic modeling to build surrogate functions to approximate the original objective function. With this procedure the use of a branch and bound technique becomes possible to solve the mixed integer problem in contrast to methods working directly on the simulation results, which impedes relaxation of integer variables. We present promising preliminary numerical results on the benchmarking problem and point the way towards improvement and future work.