The Solution of Groundwater Quality Management Problems with a Nonconvex Feasible Region Using a Cutting Plane Optimization Technique

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Abstract
In groundwater quality management problems the concentration constraints have a nonlinear behavior which may be described either as a convex or a nonconvex function. Therefore the feasible region, which is defined as the intersection of all of these constraints, can be either a convex or a nonconvex set. A review of existing optimization algorithms for the solution of the groundwater quality management problem indicates that the majority of them have an inability to determine a global optimum when nonconvexity occurs. In an earlier paper that appeared in this journal [Karatzas and Finder, 1993], the outer approximation method, a global optimization technique, was presented for the solution of groundwater management problems with convex constraints. The problem was formulated to minimize a concave objective function over a compact convex set of constraints. In the present study the same concept is applied to problems with nonconvex constraints. While the main concept of the current approach remains the same as that in our earlier study, there is a significant difference in the determination of the cutting hyperplane. The nonconvexity of the domain requires a special approach to insure that the introduction of the cutting hyperplane does not eliminate any part of the nonconvex feasible region. In this work the theory of the developed algorithm is presented and subsequently applied to two groundwater quality problems. In the first example a hypothetical aquifer is considered to illustrate the performance of the methodology. In the second example a groundwater quality management problem in Woburn, Massachusetts, is solved. Results obtained are compared with those generated by MINOS 5.1.