Finding the densest subgraph (DS) from a graph is a fundamental problem in graph databases. The DS obtained, which reveals closely related entities, has been found to be useful in various application domains such as e-commerce, social science, and biology. However, in a big graph that contains billions of edges, it is desirable to find more than one subgraph cluster that is not necessarily the densest, yet they reveal closely related vertices. In this paper, we study the locally densest subgraph (LDS), a recently proposed variant of DS. An LDS is a subgraph which is the densest among the ``local neighbors''. Given a graph $G$, a number of LDSs can be returned, which reflect different dense regions of $G$ and thus give more information than DS. The existing LDS solution suffers from low efficiency. We thus develop a convex-programming-based solution that enables powerful pruning. We also extend our algorithm to triangle-based density to solve LTDS problem. Based on current algorithms, we propose a unified framework for the LDS and LTDS problems. Extensive experiments on thirteen real large graph datasets show that our proposed algorithm is up to four orders of magnitude faster than the state-of-the-art.