Derivative-Free Optimization Methods for Handling Fixed Costs in Optimal Groundwater Remediation Design
Creators: name: Hemker, Thomas
name: Fowler, Kathleen
name: Hemker, Thomas
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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.