论文标题
近端增强拉格朗日方法的迭代复杂性，用于解决非凸复合优化问题，并具有非线性凸约束
Iteration-complexity of a proximal augmented Lagrangian method for solving nonconvex composite optimization problems with nonlinear convex constraints
论文作者
论文摘要
本文提出并分析了一种近端增强拉格朗日（NL-iapial）方法，用于解决非线性$ \ cal k $ -convex约束的平滑非convex复合优化问题，即，与封闭的convex锥$ \ cal k $给定的订单相对于限制的限制。每种NL-ia-apial迭代均包括不确定地通过加速的复合梯度（ACG）方法解决近端增强拉格朗日子问题，然后进行Lagrange乘数更新。在某些温和的假设下，表明NL-iapial在$ {\ cal o}中生成了约束问题的近似固定解决方案（\ log（1/ρ）/ρ^{3}）$内部迭代，其中$ρ> 0 $是给定的容忍度。还提供了数值实验来说明所提出方法的计算效率。
This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear $\cal K$-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone $\cal K$. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite gradient (ACG) method followed by a Lagrange multiplier update. Under some mild assumptions, it is shown that NL-IAPIAL generates an approximate stationary solution of the constrained problem in ${\cal O}(\log(1/ρ)/ρ^{3})$ inner iterations, where $ρ>0$ is a given tolerance. Numerical experiments are also given to illustrate the computational efficiency of the proposed method.
