论文标题
迭代近端加速拉格朗日方法的迭代复杂性，用于求解线性约束的平滑非凸复合优化问题
Iteration-complexity of an inexact proximal accelerated augmented Lagrangian method for solving linearly constrained smooth nonconvex composite optimization problems
论文作者
论文摘要
本文提出并建立了不精确的近端加速拉格朗日（IPAAL）方法的迭代复杂性，以解决线性约束的平滑非凸复合综合优化问题。每次iPaal迭代都包括不确定地通过加速的复合梯度（ACG）方法求解近端增强拉格朗日子问题，然后进行合适的Lagrange乘数更新。结果表明，iPaal最多可以在$ {\ cal o}（\ log（1/ρ）/ρ^{3}）$ acg迭代中生成近似的固定解决方案，其中$ρ> 0 $是给定的公差。还表明，在额外强度更强的假设下，可以将先前的复杂性结合到$ {\ cal o}（\ log（1/ρ）/ρ^{2.5}）$。假设初始点既不是可行的，也不是目标函数的复合项的域，则可以得出上述边界。提出了一些初步的数值结果，以说明iPaal方法的性能。
This paper proposes and establishes the iteration-complexity of an inexact proximal accelerated augmented Lagrangian (IPAAL) method for solving linearly constrained smooth nonconvex composite optimization problems. Each IPAAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a suitable Lagrange multiplier update. It is shown that IPAAL generates an approximate stationary solution in at most ${\cal O}(\log(1/ρ)/ρ^{3})$ ACG iterations, where $ρ>0$ is the given tolerance. It is also shown that the previous complexity bound can be sharpened to ${\cal O}(\log(1/ρ)/ρ^{2.5})$ under additional mildly stronger assumptions. The above bounds are derived assuming that the initial point is neither feasible nor the domain of the composite term of the objective function is bounded. Some preliminary numerical results are presented to illustrate the performance of the IPAAL method.
