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从常步长梯度方法的视角看不可微凸优化增广Lagrange方法的收敛性

田朝薇1   张立卫1,*   

  1. 1. 华侨大学数学科学学院, 福建泉州 362021
  • 收稿日期:2016-05-11 出版日期:2017-03-15 发布日期:2017-03-15
  • 通讯作者: 张立卫 lwzhang@dlut.edu.cn
  • 基金资助:

    国家自然科学基金(Nos. 91330206, 11571059), 福建省中青年教师教育科研项目(No. JAT160024)

A look at the convergence of the augmented  Lagrange method for nondifferentiable convex programming from  the view of a gradient method with constant stepsize

TIAN Zhaowei1  ZHANG Liwei1,*   

  1. 1. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian, China
  • Received:2016-05-11 Online:2017-03-15 Published:2017-03-15

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摘要：

增广Lagrange方法是求解非线性规划的一种有效方法. 从一新的角度证明不等式约束非线性非光滑凸优化问题的增广Lagrange方法的收敛性. 用常步长梯度法的收敛性定理证明基于增广Lagrange函数的对偶问题的常步长梯度方法的收敛性, 由此得到增广Lagrange方法乘子迭代的全局收敛性.

关键词: 梯度法, Moreau包络, 增广Lagrange对偶, 全局收敛

Abstract:

The augmented Lagrange method is an effective method for solving nonlinear optimization problems. This paper, from a new pointview, studies the convergence of the augmented Lagrange method for the nonlinear nonsmooth convex programming problem with inequality constraints. The convergence of the gradient method with constant stepsize for the dual problem, based on the augmented Lagrange function, is demonstrated by using a convergence theorem of a gradient method with constant stepsize, from which the global convergence of the multiplier iteration of augmented Lagrange method is obtained.

Key words: gradient method, Moreau envelope, augmented Lagrangian dual, global convergence