A distributed nonsmooth convex optimization problem subject to a general type of constraint, including equality and inequality as well as bounded constraints, is studied in this paper for a multiagent network with a fixed and connected communication topology. To collectively solve such a complex optimization problem, primal–dual dynamics with projection operation are investigated under optimal conditions. For the nonsmooth convex optimization problem, a framework under the LaSalle’s invariance principle from nonsmooth analysis is established, where the asymptotic stability of the primal–dual dynamics at an optimal solution is guaranteed. For the case where inequality and bounded constraints are not involved and the objective function is twice differentiable and strongly convex, the globally exponential convergence of the primal–dual dynamics is established. Finally, two simulations are provided to verify and visualize the theoretical results.

中文翻译：

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分布式约束非光滑凸优化的投影原始动力学


本文针对具有固定且连接的通信拓扑的多智能体网络，研究了受一般类型约束（包括等式和不等式以及有界约束）约束的分布式非光滑凸优化问题。为了共同解决这样一个复杂的优化问题，在最优条件下研究了具有投影运算的原对偶动力学。针对非光滑凸优化问题，建立了非光滑分析拉萨尔不变性原理下的框架，保证了原-对偶动力学在最优解处的渐近稳定性。对于不涉及不等式和有界约束且目标函数二次可微且强凸的情况，原-对偶动力学的全局指数收敛成立。最后，提供了两个模拟来验证和可视化理论结果。