浅谈增广Lagrange方法中的二阶分析

Discussion on second-order analysis in augmented Lagrange method

从极大化基于增广Lagrange函数的对偶函数的角度,可将增广Lagrange方法的乘子的迭代解释为常步长的梯度方法。增广Lagrange方法的有效性可以通过分析对偶函数的二阶微分得到。给出等式约束优化问题和一般约束非线性规划问题的对偶函数的二阶微分估计,解释为什么常步长的梯度方法具有快的收敛速度。

From the point of view of maximizing the dual function based on augmented Lagrange function,the update of multiplier of augmented Lagrange method can be interpreted as a constant step gradient method.The effectiveness of augmented Lagrange method can be obtained by analyzing the second-order differential of dual function.In this paper,the second-order differentials of dual function for equality constrained optimization problem and general constrained nonlinear programming problem are estimated,which explains why the gradient method with constant step size has fast rate of convergence.