Exact Mixed-integer Convex Programming Formulation for Optimal Water Network Design

TL;DR

This paper presents an exact mixed-integer convex programming formulation for water network design, enabling globally optimal solutions by reformulating the nonconvex problem as an MICP and developing a specialized optimization algorithm.

Contribution

The paper introduces a novel exact MICP formulation for water network design, bridging the gap between relaxations and true nonconvex physics, and proposes a global optimization algorithm.

Findings

The exact MICP formulation outperforms relaxation-based methods on benchmark instances.

The proposed algorithm achieves globally optimal solutions efficiently.

The reformulation ensures physical feasibility of the solutions.

Abstract

In this paper, we consider the canonical water network design problem, which contains nonconvex potential loss functions and discrete resistance choices with varying costs. Traditionally, to resolve the nonconvexities of this problem, relaxations of the potential loss constraints have been applied to yield a more tractable mixed-integer convex program (MICP). However, design solutions to these relaxed problems may not be feasible with respect to the full nonconvex physics. In this paper, it is shown that, in fact, the original mixed-integer nonconvex program can be reformulated exactly as an MICP. Beginning with a convex program previously used for proving nonlinear network design feasibility, strong duality is invoked to construct a novel, convex primal-dual system embedding all physical constraints. This convex system is then augmented to form an exact MICP formulation of the original design problem. Using this novel MICP as a foundation, a global optimization algorithm is developed, leveraging heuristics, outer approximations, and feasibility cutting planes for infeasible designs. Finally, the algorithm is compared against the previous relaxation-based state of the art in water network design over a number of standard benchmark instances from the literature.