AI 中文总结
该研究针对带凸二次成本与共享仿射约束的N参与者单调线性二次广义纳什均衡问题，通过将其转化为二次规划，提出两种加速算法，实验表明所提方法性能优于现有方法。
AI 中文摘要
我们考虑N个参与者之间的广义纳什均衡问题，参与者具有凸二次成本和共享仿射约束，仅假设该博弈的伪梯度是单调的。我们证明，计算变分广义纳什均衡（v-GNE）等价于求解从参与者联合Karush-Kuhn-Tucker条件导出的单个凸二次规划（QP）。在此基础上，我们证明对该QP进行正则化可得到ε-近似v-GNE，其次优性随正则化参数线性消失。接下来，我们提出加速近端点方案和加速投影梯度法，二者在第k次迭代时均能达到O(1/k²)近似v-GNE。我们还证明，博弈的可逆雅可比矩阵可将问题降维为低维QP。理论分析和数值实验表明，所提方法在求解单调线性二次v-GNE问题时显著优于现有方法。
英文摘要
We consider generalized Nash equilibrium problems among $N$ players with convex quadratic costs and shared affine constraints, assuming only that the game's pseudogradient is merely monotone. We show that computing a variational generalized Nash equilibrium (v-GNE) is equivalent to solving a single convex quadratic program (QP) derived from the players' joint Karush--Kuhn--Tucker conditions. Building on this, we show that the regularization of such a QP yields an $\varepsilon$-approximated v-GNE with suboptimality vanishing linearly in the regularization parameter. Next, we propose an accelerated proximal-point scheme and an accelerated projected-gradient method, both attaining an $\mathcal O(1/k^2)$-approximated v-GNE at the $k$-th iteration. We also demonstrate that an invertible Jacobian of the game allows for reduction to a lower-dimensional QP. Theoretical analysis and numerical experiments show the proposed methods substantially outperform the existing approaches to solve monotone linear-quadratic v-GNE problems.
Comments23 pages, 1 figure
