Enhanced lower bounds for the global optimization of water distribution networks

Abstract

In this paper we address a global optimization approach to the problem of designing a water distribution network that satisfies specified flow demands at stated pressure head requirements. The nonlinear, nonconvex network problem is transformed into the space of certain design variables. By relaxing the nonlinear constraints in the transformed space via suitable polyhedral outer approximations, we derive a linear lower bounding problem. This problem provides an enhancement of Eiger et al.'s [1994] lower bounding scheme and takes advantage of the monotone concave-convex nature of the nonlinear constraints. Upper bounds are computed by solving a projected linear program that uses the flow conserving solution generated by the lower bounding problem. These bounding strategies are embedded within a branch-and-bound algorithm. The partitioning scheme employed reduces the gap from optimality, inducing a convergent process to a feasible solution that lies within any prescribed accuracy tolerance of global optimality. The approach is illustrated using two standard test problems from the literature. For the larger (Hanoi network) problem a better incumbent than previously reported in the literature is obtained and is proven to be within 0.486% of optimality over the specified feasible domain used for this problem.