Solving rectilinear planar location problems with barriers by a polynomial partitioning

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a nonconvex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper, the nonconvex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.