Dualization of Generalized Equations of Maximal Monotone Type
Dualization of Generalized Equations of Maximal Monotone Type
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DOI：
10.1137/s1052623498340448
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发表时间：
1999-07
期刊：
影响因子：
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通讯作者：
T. Pennanen
中科院分区：
文献类型：
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作者：
T. Pennanen
本文发展了一个简单的对偶框架，定义的广义方程的集值映射从一个线性空间到另一个。原问题涉及到两个类似形式的辅助问题，分别对应于凸规划理论中的拉格朗日问题和对偶问题。在凸规划中，替代公式可以用来获得关于给定问题的信息，然后用于数值求解。特别是，二元论可以用来间接地通过考虑一个替代配方推导出一个给定的问题的存在性标准。同样，一个给定的问题通常可以通过“对偶方法”更容易地解决。“本文最强的结果涉及单调映射。在这种背景下，对偶框架产生了几个新的准则的极大单调的复合映射。这些结果在理论上和数值求解广义方程时都是有用的。对偶框架也可以用于问题分解，因为对偶化可以导致算子分裂方法和其他特殊方法可以应用的重新表述。
This paper develops a simple duality framework for generalized equations defined by set-valued mappings from a linear space to another. The original problem is related to two auxiliary problems of the similar form, corresponding to Lagrangian and dual problems in the theory of convex programming. As in convex programming, the alternative formulations can be used to obtain information about a given problem and then used to solve it numerically. In particular, dualization can be used in deriving existence criteria for a given problem indirectly by considering one of the alternative formulations. Also, a given problem can often be solved more easily by way of a "dual method." The strongest results of this paper concern monotone mappings. In this context, the duality framework yields several new criteria for maximal monotonicity of composite mappings. These results are useful theoretically as well as in numerical solution of generalized equations. The duality framework can also be used in problem decomposition since dualization can lead to reformulations to which operator-splitting methods and other special methods can be applied.