发表机构
Tokyo Metropolitan University; Nagoya Institute of Technology; Kyoto University（东京都立大学; 名古屋工业大学; 京都大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种Bregman惯性正则化外梯度方法求解双层变分不等式问题，在偶空间进行惯性外推，推导了广义间隙函数的非渐近收敛界，并通过交通网络和多投资组合纳什均衡实验验证了有效性。
AI 中文摘要
本文考虑双层变分不等式问题，其中可行集是另一个变分不等式的解集。我们提出了一种Bregman惯性正则化外梯度方法来解决这些问题。通过在偶空间中进行惯性外推，所提方法使惯性步与Bregman三点恒等式的结构保持一致。我们还推导了广义间隙函数的显式非渐近界。更具体地，利用递减正则化，我们建立了最优性间隙的收敛率为$\mathcal{O}(1/k^{1-b})$，可行性间隙的收敛率为$\mathcal{O}(1/k^b)$，其中$0<b<1$。对于常数正则化参数$\eta>0$，我们建立了最优性间隙的$\mathcal{O}(1/k)$收敛率，以及可行性间隙的$\mathcal{O}(1/k)+\mathcal{O}(\eta)$界。此外，在交通网络和多投资组合纳什均衡问题上的数值实验证明了所提框架的实际有效性。
英文摘要
In this paper, we consider bilevel variational inequality problems, where the feasible set is the solution set of another variational inequality. We propose a Bregman inertial regularized extragradient method for solving these problems. By performing the inertial extrapolation in the dual space, the proposed method aligns the inertial step with the structure of the Bregman three-point identity. We also derive explicit non-asymptotic bounds for the generalized gap function. More precisely, using diminishing regularization, we establish convergence rates of $\mathcal{O}(1/k^{1-b})$ for the optimality gap and $\mathcal{O}(1/k^b)$ for the feasibility gap with $ 0 < b< 1$. For constant regularization parameter $η>0$, we establish an $\mathcal{O}(1/k)$ rate for the optimality gap and an $\mathcal{O}(1/k)+\mathcal{O}(η)$ bound for the feasibility gap. Furthermore, numerical experiments on traffic networks and multi-portfolio Nash equilibrium problems demonstrate the practical effectiveness of the proposed framework.
