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Γ-robust linear complementarity problems
Optimization Methods & Software ( IF 2.2 ) Pub Date : 2020-10-14 , DOI: 10.1080/10556788.2020.1825708
Vanessa Krebs 1, 2 , Martin Schmidt 3
Affiliation  

Complementarity problems are often used to compute equilibria made up of specifically coordinated solutions of different optimization problems. Specific examples are game-theoretic settings like the bimatrix game or energy market models like for electricity or natural gas. While optimization under uncertainties is rather well-developed, the field of equilibrium models represented by complementarity problems under uncertainty – especially using the concepts of robust optimization – is still in its infancy. In this paper, we extend the theory of strictly robust linear complementarity problems (LCPs) to Γ-robust settings, where existence of worst-case-hedged equilibria cannot be guaranteed. Thus, we study the minimization of the worst-case gap function of Γ-robust counterparts of LCPs. For box and 1-norm uncertainty sets we derive tractable convex counterparts for monotone LCPs and study their feasibility as well as the existence and uniqueness of solutions. To this end, we consider uncertainties in the vector and in the matrix defining the LCP. We additionally study so-called ρ-robust solutions, i.e. solutions of relaxed uncertain LCPs. Finally, we illustrate the Γ-robust concept applied to LCPs in the light of the above mentioned classical examples of bimatrix games and market equilibrium modelling.



中文翻译:

Γ-鲁棒线性互补问题

互补问题通常用于计算由不同优化问题的特定协调解决方案组成的平衡。具体的例子是博弈论设置,如双矩阵博弈或能源市场模型,如电力或天然气。虽然不确定性下的优化相当发达,但以不确定性下的互补性问题为代表的均衡模型领域——尤其是使用鲁棒优化的概念——仍处于起步阶段。在本文中,我们将严格稳健线性互补问题 (LCP) 的理论扩展到 Γ 稳健设置,其中无法保证存在最坏情况对冲均衡。因此,我们研究了 LCP 的 Γ-robust 对应物的最坏情况间隙函数的最小化。对于盒子和1-范数不确定性集,我们为单调 LCP 推导出易处理的凸对应物,并研究它们的可行性以及解决方案的存在性和唯一性。为此,我们考虑了向量和定义 LCP 的矩阵中的不确定性。我们还研究了所谓的ρ稳健解,即松弛不确定 LCP 的解。最后,我们根据上述双矩阵博弈和市场均衡建模的经典例子来说明应用于 LCP 的 Γ-robust 概念。

更新日期:2020-10-14
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