  Prescribed-time convergence mechanism has become a prominent research focus
in the current field of optimization and control due to its ability to
precisely control the target completion time. The recently arisen
prescribed-time algorithms for distributed optimization, currently necessitate
multi-stage structures to achieve global convergence. This paper introduces two
modified zero-gradient-sum algorithms, each based on a multi-stage and a
single-stage structural frameworks established in this work. These algorithms
are designed to achieve prescribed-time convergence and relax two common yet
stringent conditions. This work also bridges the gap in current research on
single-stage structured PTDO algorithm. The excellent convergence performance
of the proposed algorithms is validated through a case study.
