Deployment of small cell base stations (SBSs) overlaying the coverage area of a macrocell BS (MBS) results in a two-tier hierarchical small cell network. Cross-tier and inter-tier interference not only jeopardize primary macrocell communication but also limit the spectral efficiency of small cell communication. This paper focuses on distributed interference management for downlink small cell networks. We address the optimization of transmit strategies from both the game theoretical and the network utility maximization (NUM) perspectives and show that they can be unified in a generalized Nash equilibrium problem (GNEP) framework. Specifically, the small cell network design is first formulated as a GNEP, where the SBSs and MBS compete for the spectral resources by maximizing their own rates while satisfying global quality of service (QoS) constraints. We analyze the GNEP via variational inequality theory and propose distributed algorithms, which only require the broadcasting of some pricing information, to achieve a generalized Nash equilibrium (GNE). Then, we also consider a nonconvex NUM problem that aims to maximize the sum rate of all BSs subject to global QoS constraints. We establish the connection between the NUM problem and a penalized GNEP and show that its stationary solution can be obtained via a fixed point iteration of the GNE. We propose GNEP-based distributed algorithms that achieve a stationary solution of the NUM problem at the expense of additional signaling overhead and complexity. The convergence of the proposed algorithms is proved and guaranteed for properly chosen algorithm parameters. The proposed GNEP framework can scale from a QoS constrained game to a NUM design for small cell networks by trading off signaling overhead and complexity. ; Comment: Accepted by IEEE Journal on Selected Areas in Communications