|Low earth orbit (LEO) satellite broadband network is a crucial part of the space information network. LEO satellite constellation design is a top-level design, which plays a decisive role in the overall performance of the LEO satellite network. However, the existing works on constellation design mainly focus on the coverage criterion and rarely take network performance into the design process. In this article, we develop a unified framework for constellation optimization design in LEO satellite broadband networks. Several design criteria including network performance and coverage capability are combined into the design process. Firstly, the quality of service (QoS) metrics is presented to evaluate the performance of the LEO satellite broadband network. Also, we propose a network stability model for the rapid change of the satellite network topology. Besides, a mathematical model of constellation optimization design is formulated by considering the network cost-efficiency and stability. Then, an optimization algorithm based on non-dominated sorting genetic algorithm-Ⅱ (NSGA-Ⅱ) is provided for the problem of constellation design. Finally, the proposed method is further evaluated through numerical simulations. Simulation results validate the proposed method and show that it is an efficient and effective approach for solving the problem of constellation design in LEO satellite broadband networks.