多阶段伪单调随机变分不等式的可解性
Solvability of Multistage Pseudomonotone Stochastic Variational Inequalities
摘要：
研究了多级伪单调随机变分不等式（SVI）的可解性。一方面，一些已知的伪单调确定性变分不等式的可解性结果不能直接推广到伪单调SVI，因此我们构造了伪单调SVI与变分不等式之间的同构，并通过这种同构建立了伪单调SVI解集的存在性、凸性、有界性和紧性的理论结果.另一方面，渐进对冲算法（PHA）是解决单调SVI的一种重要而有效的方法，但它不能直接用于解决非单调SVI。我们提出了伪单调SVIs可诱导性的若干充分条件，为应用诱导PHA求解伪单调SVIs打开了大门。通过求解一个伪单调两阶段随机市场最优化问题和随机生成的两阶段伪单调线性互补性问题的数值结果，验证了该方法求解伪单调SVI的有效性.
英文摘要：
This paper focuses on the solvability of multistage pseudomonotone stochastic variational inequalities (SVIs). On one hand, some known solvability results of pseudomonotone deterministic variational inequalities cannot be directly extended to pseudomonotone SVIs, so we construct the isomorphism between both and then establish theoretical results on the existence, convexity, boundedness and compactness of the solution set for pseudomonotone SVIs via such an isomorphism. On the other hand, the progressive hedging algorithm (PHA) is an important and effective method for solving monotone SVIs, but it cannot be directly used to solve nonmonotone SVIs. We propose some sufficient conditions on the elicitability of pseudomonotone SVIs, which opens the door for applying elicited PHA to solve pseudomonotone SVIs. Numerical results on solving a pseudomonotone two-stage stochastic market optimization problem and randomly generated two stage pseudomonotone linear complementarity problems are presented to show the efficiency of the elicited PHA for solving pseudomonotone SVIs.
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