We consider a cutting-plane algorithm for solving mixed-integer semidefinite optimization (MISDO) problems. In this algorithm, the positive semidefinite (psd) constraint is relaxed, and the resultant mixed-integer linear optimization problem is solved repeatedly, imposing at each iteration a valid inequality for the psd constraint. We prove the convergence properties of the algorithm. Moreover, to speed up the computation, we devise a branch-and-cut algorithm, in which valid inequalities are dynamically added during a branch-and-bound procedure. We test the computational performance of our cutting-plane and branch-and-cut algorithms for three types of MISDO problem: random instances, computing restricted isometry constants, and robust truss topology design. Our experimental results demonstrate that, for many problem instances, our branch-and-cut algorithm delivered superior performance compared with general-purpose MISDO solvers in terms of computational efficiency and stability.