Interpretation of the dual program for optimal groundwater hydraulic control

Optimization formulations for hydraulic control that take the form of linear programs possess a corresponding dual linear program. The economic and physical interpretations of the dual linear program are examined for formulations in which hydraulic head in groundwater systems is constiained. In each case it is shown that the dual linear program has a physically meaningful interpretation. For a hydraulic gradient control formulation used for remedial analysis it is shown that the dual variable can be interpreted as the remedial benefit due to each gradient control constraint. The dual linear program maximizes the remedial benefit. The value of the dual variable can be used to compute such useful properties as the total remedial benefit of pumping at a specific location. For a formulation that optimizes aquifer yield while constraining drawdown the dual variable can be used to measure the total cost of drawdown capacity consumption per unit of pumping at a specific location. The dual program minimizes the cost of drawdown capacity consumption. By examining the meaning of the dual linear program an alternate statement of the problem under study is revealed. Quantities arising from the dual program add to the value of the optimization approach. Significant new information can be derived from existing linear optimization formulations with minimal additional computational effort

A new interior-point boundary projection method for solving nonlinear groundwater pollution control problems

A new interior-point algorithm for solving the groundwater-pollution-control design problem is presented. The algorithm requires that the objective function is differentiable in the interior region. For minimization problems with nonlinear constraints and a concave objective function, the technique is shown to be similar to an active set gradient-projection method, where the tangent of the boundary between feasible and infeasible solutions is used to determine a search direction. In this new method, however, the search direction is translated into the interior space of the feasible region. This process allows progress to be made toward improving the objective function while remaining in the feasible space and ultimately converges to a stationary point. Although the solution technique was developed to solve a groundwater control formulation with a linear objective function and nonlinear constraints, the method has been successfully applied to an unconstrained nonconcave/nonconvex formulation and may be applicable to a wide variety of problems

Presence of nonconvexity in groundwater concentration response functions

This study examines the presence of nonconvexity and multiple extrema in the feasible region defined by the concentration response function. The response function is defined in the context of concentration constraints imposed on groundwater quality management models and is constructed assuming that concentration constraints are imposed at every node in a numerical grid. The groundwater system is represented by a homogeneous numerical simulation model. The response function is graphically depicted as a function of pumping rates at two wells. The dependence of the response function on simulation model and management model characteristics is examined by varying dimensionless parameter values. This study shows that the response surface is nonconvex and contains multiple local extrema over a wide range of parameter values. These features of the surface are most prominent at the transitions from extraction to injection pumping, Binding constraints often occur at or near low-velocity zones