一类半正定变分不等式的随机下降算法

The Stochastic Descent Algorithm for a Kind of Semidefinite Variational Inequality Problem

校正投影收缩算法的下降量证明中多次使用了放大不等式,因此本文利用满足固定均值的随机数适当扩张步长,得到了一类半正定变分不等式问题的随机下降算法.在适当的假设条件下,利用马尔可夫不等式和依概率收敛的性质,给出了随机下降算法的依概率收敛性证明.通过一系列的数值试验验证了随机下降算法的有效性,并且表明了合理选择随机数的均值和方差可以提高随机下降算法的计算效率.

The amplification inequality is used for many times in the proof of drop function of correction projection and contraction algorithm, so we propose the stochastic descent algorithm for a class of semidifinite variational inequality prob- lem through the random steplength extension with the random number series satisfying the Gaussian distribution or Uni- form distribution and these random number series have a fixed mean. Subsequently, the probability convergence of sto- chastic descent algorithm is provided by the properties of Markov＇s inequality and probability convergence under some suitable conditions. Finally, some numerical experiments show the effectiveness and efficiency of the stochastic descent algorithm, and reasonable selecting mean and variance of random number can improve the efficiency of the algorithm.