A Primal Method for the Solution of the Groundwater Quality Management Problem

Abstract

A new algorithm for the solution of the groundwater quality management problem is presented. The method assumes that most of the computational effort in such problems involves evaluation of the concentrations and their derivatives with respect to the pumping rates at the control points. The methodology proposed herein is a combination of the cutting plane method and the primal method. In this approach each line search moves from a feasible point towards the solution of a subproblem with linear constraints and the same objective function as the original problem. Sensitivity analysis is used to compute the derivative of a single concentration with respect to all the pumping rates by solving only one linear system at each time step. The method has been tested with analytical convex problems and a model example. A comparison of the algorithm's performance was conducted using MINOS 5.1. For the case of a linear cost function the method shows a computational growth rate almost linearly proportional to the number of decision variables.

A New Interior-Point Boundary Projection Method For Solving Nonlinear Groundwater Pollution Control Problems

“…For example, a combined cutting-plane and primal method (Tucciarelli et al 1998) demonstrated better convergence properties than an implementation using a reduced gradient method (Murtagh and Saunders 1987) when solving a groundwater quality management formulation. The iterative steps of the combined cutting-plane and primal method result in similar behavior to that described above for active set gradient-projection methods.…”

“…The example is the same problem presented by Tucciarelli et al (1998), which is shown in Figure 4 and is stated as x max 2 and the initial value for w = 1 0. Additionally, set the termination criterion on w to 0.01; that is, when the value of w falls below this criterion the algorithm stops and the current solution is returned.…”