基于差分包络平滑的凸差分规划算法

Kaizhao Sun, X. Sun
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引用次数: 8

摘要

在本文中,我们考虑了有线性等式约束和没有线性等式约束的凸差函数的最小化问题。我们首先研究了一般DC函数的光滑近似,称为莫罗包络差(DME)平滑,其中DC函数的两个组成部分都被各自的莫罗包络所取代。所得到的光滑近似被证明是Lipschitz可微的,捕获了原始DC函数的平稳点,局部和全局最小值,并且对于广泛的DC函数具有一些增长条件,例如水平有界性和矫顽力。对于无线性约束的光滑DC规划,证明了经典梯度下降法和不精确变异体在极限处收敛到原始DC函数的平稳解,其速率为[公式:见文],其中K为两个分量的近端求值次数。此外,当数据中心规划被显式约束在仿射子空间时,我们将平滑技术与增广拉格朗日函数相结合,针对数据中心目标函数的不同结构,导出了增广拉格朗日方法(ALM)的两种变体,分别称为线性约束数据中心(LCDC)-ALM和复合LCDC-ALM。我们证明了两种算法在[公式:见文本]迭代中都找到了原始DC程序的ϵ-approximate平稳解。与现有的线性约束弱凸最小化方法相比,本文提出的基于alm的算法可以应用于更广泛的问题类别,其中目标包含一个非光滑的凹分量。最后,通过数值实验验证了算法的有效性。经费:本研究得到了美国国家科学基金会的部分支持[Grant ECCS1751747]。补充材料:电子伴侣可在https://doi.org/10.1287/ijoo.2022.0087上获得。
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Algorithms for Difference-of-Convex Programs Based on Difference-of-Moreau-Envelopes Smoothing
In this paper, we consider minimization of a difference-of-convex (DC) function with and without linear equality constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. For a smoothed DC program without linear constraints, it is shown that the classic gradient descent method and an inexact variant converge to a stationary solution of the original DC function in the limit with a rate of [Formula: see text], where K is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named linearly constrained DC (LCDC)-ALM and composite LCDC-ALM, targeting on different structures of the DC objective function. We show that both algorithms find an ϵ-approximate stationary solution of the original DC program in [Formula: see text] iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms. Funding: This work was partially supported by the NSF [Grant ECCS1751747]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoo.2022.0087 .
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