不确定多智能体最优化问题解的概率可行性
On the probabilistic feasibility of solutions in multi-agent optimization problems under uncertainty
摘要：
我们研究了两类不同的不确定多智能体优化方案的随机解的概率可行性。我们首先假设只有规划的约束条件受到不确定性的影响，而费用函数是任意的。利用最近的后验事态发展的情况下的方法，我们提供概率保证所有的可行解的方案正在研究中。这个结果是特别有用的情况下，数值困难的收敛性的解决方案寻求算法阻碍了精确量化的最优解。此外，它可以应用于代理人的激励导致次优解决方案的情况，例如，在不合作的情况下。然后，我们专注于优化程序的费用函数承认一个聚合表示，并依赖于不确定性，而约束是确定性的。通过利用所研究的程序的结构并利用所谓的支持度等级概念，我们为最优解提供了与代理无关的鲁棒性证书，即，所构造的约束违反概率界不依赖于主体的数量，而仅依赖于主体决策的维数。随着代理数量的增加，这大大减少了实现一定水平的概率鲁棒性所需的样本的样本个数。本文提供的所有健壮性证书都是免费分发的，可以与任何优化算法一起使用。我们的理论结果是伴随着一个数字的案例研究，涉及一个车队的电动汽车的充电控制问题。
英文摘要：
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 a posteriori developments of the scenario approach, we provide probabilistic guarantees for all feasible solutions of the program under study. This result is particularly useful in cases where numerical difficulties related to the convergence of the solution-seeking algorithm hinder the exact quantification of the optimal solution. Furthermore, it can be applied to cases where the agents' incentives lead to a suboptimal solution, e.g., under a non-cooperative setting. We then focus on optimization programs where the cost function admits an aggregate representation and depends on uncertainty while constraints are deterministic. 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 the agents' decision. This substantially reduces the number of samples required to achieve a certain level of probabilistic robustness as the number of agents increases. 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 involving a charging control problem of a fleet of electric vehicles.
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