Iteration Complexity of an Inner Accelerated Inexact Proximal Augmented Lagrangian Method Based on the Classical Lagrangian Function
本文研究了一种内加速非精确近端增广拉格朗日方法，用于求解线性约束光滑非凸复合优化问题，证明了其达到近似稳定解所需的迭代次数上界，并给出了数值结果。
.This paper establishes the iteration complexity of an inner accelerated inexact proximal augmented Lagrangian (IAIPAL) method for solving linearly constrained smooth nonconvex composite optimization problems that is based on the classical augmented Lagrangian (AL) function. More specifically, each IAIPAL iteration consists of inexactly solving a proximal AL subproblem by an accelerated composite gradient (ACG) method followed by a classical Lagrange multiplier update. Under the assumption that the domain of the composite function is bounded and the problem has a Slater point, it is shown that IAIPAL generates an approximate stationary solution in \({\mathcal O}(\varepsilon ^{-5/2}\log^2 \varepsilon ^{-1})\) ACG iterations where \(\varepsilon \gt 0\) is a tolerance for both stationarity and feasibility. Moreover, the above bound is derived without assuming that the initial point is feasible. Finally, numerical results are presented to demonstrate the strong practical performance of IAIPAL.Keywordsinexact proximal augmented Lagrangian methodlinearly constrained smooth nonconvex composite programsinner accelerated first-order methodsiteration complexityMSC codes47J2249M2790C2590C2690C3090C6065K10