单调逆变分不等式的正则化投影算法
Regularized Projection Algorithms for Monotone Inverse Variational Inequalities
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中文总结 AI 辅助
针对单调随机逆变分不等式，提出结合Tikhonov正则化与递增批量大小的算法，证明迭代有界及收敛，给出期望平方残差的非渐近率，还介绍了确定性变体及数值实验。
中文摘要 AI 辅助
随机逆变分不等式（SIVIs）出现在不确定性下观测均衡响应的应用中。现有方法依赖共强制性或强单调性，而一般单调SIVIs了解较少。我们提出一种正则化投影算法，结合Tikhonov正则化与递增批量大小。在单调性和Lipschitz连续性下，证明迭代几乎必然有界及其到SIVI解集距离几乎必然收敛。还建立了一般单调性下期望平方残差的首个显式非渐近率$O(T^{-1/2})$。
英文摘要
Stochastic inverse variational inequalities (SIVIs) arise in applications in which equilibrium responses are observed under uncertainty, such as inverse road pricing and network equilibrium control. Existing methods typically rely on co-coercivity or strong monotonicity, while general monotone SIVIs remain less understood. We propose a regularized projection algorithm that combines Tikhonov regularization with increasing batch sizes. Under monotonicity and Lipschitz continuity, we prove almost sure boundedness of the iterates and almost sure convergence of their distance to the SIVI solution set. We further establish, to the best of our knowledge, the first explicit nonasymptotic rate of $O(T^{-1/2})$ for the expected squared residual under general monotonicity. This yields $O(ε^{-2})$ iterations and $O(ε^{-4-2δ})$ stochastic oracle calls, for any $δ>0$, to obtain an $ε$-solution in expected squared residual. A deterministic variant attains the same iteration complexity using $O(ε^{-2})$ exact operator evaluations. Numerical experiments illustrate the proposed methods on monotone SIVI problems.
