In this paper, we study a facility cost optimization problem in a competitive market. Our objective is to distribute an available budget to some newly opened facilities to maximize an expected captured customer demand, assuming that customers will select a facility to visit according to a random utility maximization model. In this work, given the fact that the objective function of this problem is highly non-convex and challenging to solve exactly, we propose a technique to approximate the objective function by piece-wise linear functions, making it possible to reformulate the problem as a mixed-integer linear or conic program, which can further be solved by a commercial solver such as CPLEX. We also explore an outer-approximation algorithm to solve the approximate problem. Computational results are provided to demonstrate the performances of our approaches.