Peer to peer based live multimedia streaming systems over the Internet have been gaining popularity these days because they can be easily deployed without router based IP multicast capabilities. However, the peers frequently join and leave the system so that some peers may experience service disruptions, which degrade the overall quality and reputation of the services. In this situation, a hybrid approach is considered as a viable alternative. In the hybrid approach, the service providers deploy multiple proxies all over the Internet and make them constitute a core tree in the peer to peer system so that the proxies behave as stable nodes that are always on. Thus the system quality can increase. In this paper, we propose a centralized core tree construction scheme based on various optimization algorithms. The scheme focuses on constructing a core tree that maximizes the overall quality of the system. Because the real time multimedia streaming systems have higher QoS requirements than the content distribution such as Akamai, we take into account not only the locations of the proxies but also the topology formed by the proxies. The general framework can obtain a tree for arbitrary performance metrics not only for a specific metric such as minimum delay. Through simulations over diverse scenarios, we show that our optimization based approaches can provide a better core tree than some greedy heuristic approaches. ©2010 IEEE.