This paper proposes mathematical optimization models for solving the network planning problem using millimeter wave technology for 5G wireless communications networks. To this end, it is assumed that a set of users, M={1,& mldr;,m}, and a set of base stations, N={1,& mldr;,n}, are deployed randomly in a square area. In particular, the base stations should be connected, forming a star backbone so that users can connect to their nearest active base stations forming the backbone where the connections are symmetric. In particular, the first two models maximize the number of users connected to the backbone and minimize the distance costs of connecting users to the base stations, and distances of connecting the base stations themselves. Similarly, the last two models maximize and minimize the same objectives and the number of base stations to be activated to form the star backbone. Each user is allowed to connect to a unique active base station. In general, the millimeter wave technology presents a high path loss. Consequently, the transmission distances should be no larger than 300 m at most for different radial transmissions. Thus, a direct line of sight between users and base stations is assumed. Finally, we propose local search-based algorithms that allow finding near-optimal solutions for all our tested instances. Our numerical results indicate that we can solve network instances optimally with up to k=100, n=200, and m=5000 users.