In this paper, we present two approaches for solving an $\mathcal{NP}$-hard problem: minimum maximal flow problem, i.e., $\min\{\| \xi\|\,| \, \xi \text{ is a maximal flow}\}$. We introduce lower bounds on flow, and cast the problem into a minimization of a concave function over a convex set. We solve the problem by a global optimization method. As an application, we consider the minimum maximal matching problem. Some numerical experiments are also reported.