DERIVATIVE-FREE OPTIMIZATION METHODS FOR HANDLING FIXED COSTS IN OPTIMAL GROUNDWATER REMEDIATION DESIGN

We consider a hydraulic capture application for water resources management that includes a fixed installation cost in addition to operating costs. The result is a simulationbased, 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. Such optimization tasks are not unique to subsurface management, rather efficient simulation-based methods are needed in the whole field of computational engineering. All the approaches used below do not need the gradient of the objective function, only function values for minimization. In one approach, we bypass including the number of wells as a decision variable by defining an inactive-well threshold. In another approach, we use penalty coefficients proposed in the literature to transform the discontinuous problem into a continuous one. For the two above formulations, we use the implicit filtering algorithm. In the third approach, we introduce a mixed-integer problem formulation and use an iterative stochastic modeling technique to build surrogate functions that approximate the objective function. With this new 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 numerical results on the benchmarking problem and point the way towards improvement and future work.