PPA-Like Contraction Methods for Convex Optimization: A Framework Using Variational Inequality Approach
PPA-Like Contraction Methods for Convex Optimization: A Framework Using Variational Inequality Approach
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DOI：
10.1007/s40305-015-0108-9
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发表时间：
2015-12-01
影响因子：
1.4
通讯作者：
He, Bing-Sheng
中科院分区：
文献类型：
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作者：
He, Bing-Sheng
线性约束凸优化有着广泛的应用。线性约束凸优化的一阶最优性条件是单调变分不等式（VI）。在求解VI问题中，欧氏范数下的邻近点算法（PPA）是经典而又抽象的。因此，经典的PPA只起着重要的理论作用，很少用于实际的科学计算。本文对最近发展起来的H-范数（H是正定矩阵）自定义PPA进行了综述。在定制PPA框架下，可以方便地构造出求解不同线性约束的凸优化问题的压缩型方法。在每一次迭代中，我们只需要解决的近端子问题，有封闭形式的解决方案，或可以有效地解决了高精度。一些新的应用和数值实验的报告。此外，通过使用预测-校正统一框架，将原始-对偶混合梯度法改进为收敛算法。利用变分不等式方法，该框架的压缩收敛性和收敛速度的证明更加一般和简单。
Linearly constrained convex optimization has many applications. The first-order optimal condition of the linearly constrained convex optimization is a monotone variational inequality (VI). For solving VI, the proximal point algorithm (PPA) in Euclidean-norm is classical but abstract. Hence, the classical PPAonly plays an important theoretical role and it is rarely used in the practical scientific computation. In this paper, we give a review on the recently developed customized PPA in H-norm (H is a positive definite matrix). In the frame of customized PPA, it is easy to construct the contraction-type methods for convex optimization with different linear constraints. In each iteration of the proposed methods, we need only to solve the proximal sub-problems which have the closed form solutions or can be efficiently solved up to a high precision. Some novel applications and numerical experiments are reported. Additionally, the original primal-dual hybrid gradient method is modified to a convergent algorithm by using a prediction-correction uniform framework. Using the variational inequality approach, the contractive convergence and convergence rate proofs of the framework are more general and quite simple.