Numerical methods are employed to address an inverse problem associated with a water distribution network characterized by a complex loopback structure. The problem is to ascertain the locations and magnitudes of leaks based on measurements of unsteady flow characteristics at certain points in the pipeline. Key aspects of the problem include the involvement of impulse functions within a system of hyperbolic differential equations, the lack of traditional initial conditions, and the specification of nonseparated boundary conditions between states at the endpoints of adjacent pipeline segments. The problem is transformed into a parametric optimal control problem, devoid of initial conditions but featuring nonseparated boundary conditions. The latter problem is tackled using first-order optimization methods. The paper presents the outcomes of numerical experiments. Notably, this research distinguishes itself from others by addressing the inverse problem of determining leak locations and magnitudes within an unsteady flow scenario in a water distribution network with a complex (loopback) structure, as opposed to studies focusing on steady flow or transient flow in simpler pipeline configurations.