The algebraic degree of geometric optimization problems
Discrete & Computational Geometry
| Hi-index | 0.98 |
A logarithmic potential function is used to model the noxious facility location problem and a closed-form solution (in terms of radicals) is given for locating a noxious facility within the convex hull of three cities. The location of the facility is chosen to coincide with the critical (or saddle-) points of the logarithmic potential function. Since the logarithmic potential function is harmonic, the maximum and minimum solutions will lie on the boundary of the region under consideration and will not coincide with the critical points of the function. It is shown that only in the three-city case can an exact formula be obtained for the critical points; i.e., determining the critical points within four or more cities will necessarily lead to an approximate solution. Algorithmic procedures to solve for the n-city problem, however, are available and are discussed. The geometry of the critical points associated with the logarithmic potential is examined in detail for the three-city case. Using a classical result on the geometry of complex zeros, a constructive approach is used to parameterize the conic sections associated with these points. A computer implementation of the parameterization is given, and by varying certain ''coefficients of toxicity'' associated with each city point, the behavior of the complex zeros are examined.