Distributed Free-Will Arbitrary Time Optimization for Multi-Agent Systems with Zero-Gradient-Sum Algorithm

In this article, a free-will arbitrary time distributed convex optimization problem is studied for multi-agent systems. The distributed arbitrary optimization means that control strategy not only drive all agents to reach a consensus but also cooperatively minimize the sum of the objective functions of each agent within the prespecified arbitrary time. Different from the finite-time and fixed-time optimization problems, we can prespecified arbitrary settling time. In order to solve this problem, we proposed a distributed first-order arbitrary time optimization control algorithm within Zero-Gradient-Sum(ZGS) framework, and consider the strongly convex function for each agent. Theoretical studies indicate that the proposed algorithms can minimize the team objective function and all agents also can reach a consensus within the prespecified arbitrary time. Simulation examples confirm the theoretical findings.