本文给出了求解拟变分不等式问题的一种投影算法,在算法的第二次投影步中,把到一般闭凸集上的投影松弛为到半空间的投影,这在一定程度上减少了计算的难度。该算法的全局收敛性得到证明。<br/>In this paper, we present a projection-like algorithm for solving the quasi-variational inequality problem. In the second projection step of the algorithm, we replace the orthogonal projection onto a general closed convex set with a projection onto a halfspace, which reduces the difficulty of cal-culation to some extent. The global convergence of the algorithm is given.
拟变分不等式问题在经济、工程、最优化和控制等领域都有着广泛的应用。例如经济问题中的Nash均衡问题可以等价地转化为一个变分不等式问题,而广义Nash均衡问题可以等价地转化为一个拟变分不等式问题[1] 。两者比较而言,广义Nash均衡问题更接近经济问题的实际。因此,研究拟变分不等式问题的有效数值解法有着重要的理论意义和实用价值。对于拟变分不等式问题的研究,正如文献[2] 所言:“the study of the QVI to date is in its infancy at best”。因此,寻找和设计求解拟变分不等式问题的算法是很有意义的工作。在求解拟变分不等式问题的算法方面,文献[3] 提出了两步投影法,即:每次迭代需要计算预测步和校正步;文献[4] 在假设算子具有协强制性条件下,对[3] 中的算法做了改进,并给出了收敛性分析。另外,文献[5] 也给出了投影类算法。在上述已有的投影类算法中,所有的投影都是到由当前迭代点所产生的一个闭凸集上或原问题的可行解集上。我们知道,到一般闭凸集上的投影有时不易求得或者很难计算。受文献[6] 求解变分不等式问题算法的启发,我们设计了求解拟变分不等式问题的一种投影算法,在算法的校正步中,我们把到一般闭凸集上的投影松弛为到半空间的投影,而这里构造半空间时,还成功避免了次梯度的求解,这在一定程度上减少了计算的难度。
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