On the asymptotic stability of proximal algorithms for convex optimization problems with multiple non-smooth regularizers
On the asymptotic stability of proximal algorithms for convex optimization problems with multiple non-smooth regularizers
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DOI：
10.23919/acc53348.2022.9867197
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发表时间：
2022-06
期刊：
影响因子：
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通讯作者：
Ibrahim Kurban Özaslan;Sepideh Hassan-Moghaddam;M. Jovanović
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文献类型：
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作者：
Ibrahim Kurban Özaslan;Sepideh Hassan-Moghaddam;M. Jovanović
我们考虑复合优化问题，其中的目标函数是由一个光滑的凸项和多个，潜在的不可微，凸正则化。我们表明，原始-对偶方法的基础上，近端增广拉格朗日，这是最初引入的问题，两个组件，可以直接扩展到这个多块的情况下。此外，我们证明了连续时间的原始-对偶动力学产生的近增广拉格朗日是全局渐近稳定的，即使在多块的情况下，如果平衡点集是紧凑的。这与ADMM相反，ADMM中附加的假设，例如，某些组件需要强凸性。然后，我们研究了三块问题与两个非光滑正则化，并建立了全球渐近稳定的分裂动力学产生的近端增广拉格朗日。
We consider composite optimization problems in which the objective function is given by the sum of a smooth convex term and multiple, potentially non-differentiable, convex regularizers. We show that a primal-dual method based on the proximal augmented Lagrangian, which was originally introduced for problems with two components, can be directly extended to this multi-block case. Moreover, we prove that the continuous-time primal-dual dynamics resulting from the proximal augmented Lagrangian are globally asymptotically stable even in the multi-block case if the set of equilibrium points is compact. This is in contrast to ADMM where additional assumptions, e.g., strong convexity of some components, are required. We then examine three-block problems with two non-smooth regularizers and establish global asymptotic stability of splitting dynamic resulting from the proximal augmented Lagrangian.