In this paper, a class of optimization problems is investigated, where the objective function is the sum of $N$ convex functions viewed as local functions and the constraints are $N$ nonidentical closed convex sets. Additionally, it is aimed to solve the considered optimization problem in a distributed manner and thus a sequence of time-varying unbalanced directed graphs is introduced first to depict the information connection topologies. Then, the novel push-sum based constrained optimization algorithm (PSCOA) is developed, where the new gradient descent-like method is applied to settle the involved closed convex set constraints. Furthermore, the rigorous convergence analysis is shown under some standard and common assumptions and it is proved that the developed distributed discrete-time algorithm owns a convergence rate of $\mathcal{O}\left(\frac{lnt}{\sqrt{t}}\right)$ in general case. Specially, the convergence rate of $\mathcal{O}\left(\frac{1}{t}\right)$ can be further obtained under the assumption that at least one objective function is strongly convex. Finally, simulation results are given to demonstrate the validity of the theoretical results.

在本文中，研究了一类优化问题，其中目标函数是 $N$ 凸函数被视为局部函数，约束是 $N$非相同闭凸集。此外，它旨在以分布式方式解决所考虑的优化问题，因此首先引入一系列时变不平衡有向图来描述信息连接拓扑。然后，开发了新的基于推和的约束优化算法（PSCOA），其中应用了新的类梯度下降方法来解决所涉及的封闭凸集约束。此外，在一些标准和通用假设下进行了严格的收敛分析，证明了所开发的分布式离散时间算法的收敛速度为$\mathcal{哦}\left(\frac{输入吨}{\sqrt{吨}}\right)$在一般情况下。特别地，收敛速度为$\mathcal{哦}\left(\frac{1}{吨}\right)$可以在至少一个目标函数是强凸的假设下进一步得到。最后，给出仿真结果来证明理论结果的有效性。