Lagrangian-Dual Functions and Moreau--Yosida Regularization
Lagrangian-Dual Functions and Moreau--Yosida Regularization
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DOI：
10.1137/060673746
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发表时间：
2008-02
期刊：
影响因子：
--
通讯作者：
F. Meng;G. Zhao;M. Goh;R. Souza
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文献类型：
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作者：
F. Meng;G. Zhao;M. Goh;R. Souza
本文研究了一类凸优化问题的拉格朗日对偶问题。我们首先讨论了拉格朗日对偶函数$\varphi$的半光滑性。这个性质然后被用来研究函数$\varphi$的Moreau-Yosida正则化$\eta$的二阶性质，例如，正则化函数的梯度g的半光滑性我们表明，$\varphi$和$g$是分段C^2 $和半光滑，分别为某些情况下的优化问题。我们建立了原问题和相应的拉格朗日对偶问题的正则化的Fenchel共轭之间的关系。我们还发现了一些优化问题的拉格朗日对偶函数$\varphi$是不分段光滑的情况。然而，它的正则化函数仍然具有良好的二阶性质。最后，在对偶函数的上图结构下，给出了研究梯度的半光滑性的另一种方法。
In this paper, we consider the Lagrangian-dual problem of a class of convex optimization problems. We first discuss the semismoothness of the Lagrangian-dual function $\varphi$. This property is then used to investigate the second-order properties of the Moreau-Yosida regularization $\eta$ of the function $\varphi$, e.g., the semismoothness of the gradient $g$ of the regularized function $\eta$. We show that $\varphi$ and $g$ are piecewise $C^2$ and semismooth, respectively, for certain instances of the optimization problem. We establish a relationship between the original problem and the Fenchel conjugate of the regularization of the corresponding Lagrangian dual problem. We also find some instances of the optimization problem whose Lagrangian-dual function $\varphi$ is not piecewise smooth. However, its regularized function still possesses nice second-order properties. Finally, we provide an alternative way to study the semismoothness of the gradient under the structure of the epigraph of the dual function.