Smoothing Projected Gradient Method and Its Application to Stochastic Linear Complementarity Problems
Smoothing Projected Gradient Method and Its Application to Stochastic Linear Complementarity Problems
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DOI：
10.1137/070702187
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
Chao Zhang;Xiaojun Chen
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文献类型：
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作者：
Chao Zhang;Xiaojun Chen
针对闭凸集上的极小化问题，提出了一种光滑投影梯度（SPG）方法，其中目标函数是局部Lipschitz连续但非凸不可微的.我们表明，由SPG方法产生的任何聚点是与该方法中使用的平滑函数，这是一个克拉克驻点在许多应用中的驻点。我们将SPG方法应用于随机线性互补问题（SLCP）和图像恢复问题。我们研究了由方向导数定义的不动点，并给出了SLCP的期望残差最小化（ERM）公式的局部极小的充要条件。用SPG方法求解随机产生的SLCP和大规模图像恢复问题的初步数值实验表明，SPG方法是有前途的。
A smoothing projected gradient (SPG) method is proposed for the minimization problem on a closed convex set, where the objective function is locally Lipschitz continuous but nonconvex, nondifferentiable. We show that any accumulation point generated by the SPG method is a stationary point associated with the smoothing function used in the method, which is a Clarke stationary point in many applications. We apply the SPG method to the stochastic linear complementarity problem (SLCP) and image restoration problems. We study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization (ERM) formulation of SLCP. Preliminary numerical experiments using the SPG method for solving randomly generated SLCP and image restoration problems of large sizes show that the SPG method is promising.