We investigate the competitive  facility location problem (CFLP) with partially binary choice rule where  two companies sequentially open a limited number new facilities to maximize their market shares of demand, requiring customers to patronize, for each company, at most a single facility. The problem is a bilevel mixed-integer nonlinear programming (MINLP) problem and can be easily rewritten as a single level MINLP problem. To efficiently solve the  single level MINLP problem, we provide two MILP reformulations for the CFLP based on which  the sophisticated branch-and-cut algorithm can be applied. The first one is based on a class of improved submodular cuts and is shown to be much stronger than the one based on the classic submodular cuts in terms of providing the linear programming (LP) relaxation bound. The second one can be seen as an extended formulation of the first one but has a significantly less number of constraints. We show that the LP relaxations of the two formulations are equivalent. We present computational results which demonstrate that by using our branch-and-cut approaches, instances that are considerably larger than have been considered before can be solved to optimality.