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Probabilistic feasibility guarantees for solution sets to uncertain variational inequalities
arXiv - CS - Systems and Control Pub Date : 2020-05-19 , DOI: arxiv-2005.09420
Filippo Fabiani, Kostas Margellos, Paul J. Goulart

We develop a data-driven approach to the computation of a-posteriori feasibility certificates to the solution sets of variational inequalities affected by uncertainty. Specifically, we focus on instances of variational inequalities with a deterministic mapping and an uncertain feasibility set, and represent uncertainty by means of scenarios. Building upon recent advances in the scenario approach literature, we quantify the robustness properties of the entire set of solutions of a variational inequality, with feasibility set constructed using the scenario approach, against a new unseen realization of the uncertainty. Our results extend existing results that typically impose an assumption that the solution set is a singleton and require certain non-degeneracy properties, and thereby offer probabilistic feasibility guarantees to any feasible solution. We show that assessing the violation probability of an entire set of solutions, rather than of a singleton, requires enumeration of the support constraints that "shape" this set. Additionally, we propose a general procedure to enumerate the support constraints that does not require a closed form description of the solution set, which is unlikely to be available. We show that robust game theory problems can be modelling via uncertain variational inequalities, and illustrate our theoretical results through extensive numerical simulations on a case study involving an electric vehicle charging coordination problem.

中文翻译:

不确定变分不等式的解集的概率可行性保证

我们开发了一种数据驱动的方法来计算受不确定性影响的变分不等式的解决方案集的后验可行性证明。具体而言,我们关注具有确定性映射和不确定性可行性集的变分不等式实例,并通过场景表示不确定性。基于情景方法文献的最新进展,我们量化了变分不等式的整个解决方案集的稳健性属性,使用情景方法构建了可行性集,以对抗新的未知的不确定性实现。我们的结果扩展了现有的结果,这些结果通常假设解集是单例并需要某些非简并属性,从而为任何可行的解决方案提供概率可行性保证。我们表明,评估一整套解决方案的违反概率,而不是单个解决方案,需要枚举“塑造”这个集合的支持约束。此外,我们提出了一个枚举支持约束的通用程序,它不需要解决方案集的封闭形式描述,这不太可能可用。我们展示了稳健的博弈论问题可以通过不确定的变分不等式进行建模,并通过对涉及电动汽车充电协调问题的案例研究进行广泛的数值模拟来说明我们的理论结果。需要枚举“塑造”这个集合的支持约束。此外,我们提出了一个枚举支持约束的通用程序,它不需要解决方案集的封闭形式描述,这不太可能可用。我们展示了稳健的博弈论问题可以通过不确定的变分不等式进行建模,并通过对涉及电动汽车充电协调问题的案例研究进行广泛的数值模拟来说明我们的理论结果。需要枚举“塑造”这个集合的支持约束。此外,我们提出了一个枚举支持约束的通用程序,它不需要解决方案集的封闭形式描述,这不太可能可用。我们展示了稳健的博弈论问题可以通过不确定的变分不等式进行建模,并通过对涉及电动汽车充电协调问题的案例研究进行广泛的数值模拟来说明我们的理论结果。
更新日期:2020-11-05
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