  This paper develops distributed synchronous and asynchronous algorithms for
the large-scale semi-definite programming with diagonal constraints, which has
wide applications in combination optimization, image processing and community
detection. The information of the semi-definite programming is allocated to
multiple interconnected agents such that each agent aims to find a solution by
communicating to its neighbors. Based on low-rank property of solutions and the
Burer-Monteiro factorization, we transform the original problem into a
distributed optimization problem over unit spheres to reduce variable
dimensions and ensure positive semi-definiteness without involving
semi-definite projections, which are computationally expensive. For the
distributed optimization problem, we propose distributed synchronous and
asynchronous algorithms, both of which reduce computational burden and storage
space compared with existing centralized algorithms. Specifically, the
distributed synchronous algorithm almost surely escapes strict saddle points
and converges to the set of optimal solutions to the optimization problem. In
addition, the proposed distributed asynchronous algorithm allows communication
delays and converges to the set of critical points to the optimization problem
under mild conditions. By applying proposed algorithms to image segmentation
applications, we illustrate the efficiency and convergence performance of the
two proposed algorithms.
