We investigate the probabilistic feasibility of randomized solutions to two distinct classes of uncertain multi-agent optimization programs. We first assume that only the constraints of the program are affected by uncertainty, while the cost function is arbitrary. Leveraging recent developments on a posteriori analysis within the scenario approach, we provide probabilistic guarantees for all feasible solutions of the program under study. This is particularly useful in cases where the numerical implementation of the solution-seeking algorithm prevents the exact quantification of the optimal solution. Furthermore, this result provides guarantees for the entire solution set of optimization programs with uncertain convex constraints and (possibly) non-convex cost function. We then focus on optimization programs with deterministic constraints, where the cost function depends on uncertainty and admits an aggregate representation of the agents’ decisions. By exploiting the structure of the program under study and leveraging the so called support rank notion, we provide agent-independent robustness certificates for the optimal solution, i.e., the constructed bound on the probability of constraint violation does not depend on the number of agents, but only on the dimension of each agent’s decision space. This substantially reduces the amount of samples required to achieve a certain level of probabilistic robustness for a larger number of agents. All robustness certificates provided in this paper are distribution-free and can be used alongside any optimization algorithm. Our theoretical results are accompanied by a numerical case study of a charging control problem for a fleet of electric vehicles.

我们研究了对两类不同的不确定多智能体优化程序的随机解决方案的概率可行性。我们首先假设只有程序的约束受到不确定性的影响，而成本函数是任意的。利用后验的最新发展在情景方法的分析中，我们为正在研究的程序的所有可行解决方案提供概率保证。这在解决方案寻求算法的数值实现阻止最佳解决方案的精确量化的情况下特别有用。此外，该结果为具有不确定凸约束和（可能）非凸成本函数的优化程序的整个解决方案集提供了保证。然后，我们专注于具有确定性约束的优化程序，其中成本函数取决于不确定性并承认代理决策的聚合表示。通过利用所研究程序的结构并利用所谓的支持等级概念，我们为最优解决方案提供独立于代理的鲁棒性证书，即 约束违反概率的构造边界不取决于代理的数量，而仅取决于每个代理的决策空间的维度。这大大减少了为大量代理实现一定水平的概率鲁棒性所需的样本量。本文中提供的所有稳健性证书都是无分布的，可以与任何优化算法一起使用。我们的理论结果伴随着电动汽车车队充电控制问题的数值案例研究。本文中提供的所有稳健性证书都是无分布的，可以与任何优化算法一起使用。我们的理论结果伴随着电动汽车车队充电控制问题的数值案例研究。本文中提供的所有稳健性证书都是无分布的，可以与任何优化算法一起使用。我们的理论结果伴随着电动汽车车队充电控制问题的数值案例研究。