Foundations of Gauge and Perspective Duality
Foundations of Gauge and Perspective Duality
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DOI：
10.1137/17m1119020
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发表时间：
2017-02
期刊：
影响因子：
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通讯作者：
A. Aravkin;J. Burke;D. Drusvyatskiy;M. Friedlander;Kellie J. MacPhee
中科院分区：
文献类型：
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作者：
A. Aravkin;J. Burke;D. Drusvyatskiy;M. Friedlander;Kellie J. MacPhee
我们重新审视规范对偶的基础，并证明它可以用基于摄动框架的现代对偶方法来解释。因此，我们将规范对偶和fenchell - rockafellar对偶置于同等地位，包括将规范对偶变量解释为灵敏度度量，并展示如何从规范对偶的原始解中恢复原始解。这一优势可以直接证明，规范对偶的fenchell - rockafellar对偶的最优解正是由最优值重新缩放的原始解。我们将规范对偶框架扩展到功能分量为一般非负凸函数的情况，包括分段线性二次函数和由回归中使用的广义线性模型产生的约束的问题。
We revisit the foundations of gauge duality and demonstrate that it can be explained using a modern approach to duality based on a perturbation framework. We therefore put gauge duality and Fenchel-Rockafellar duality on equal footing, including explaining gauge dual variables as sensitivity measures, and showing how to recover primal solutions from those of the gauge dual. This vantage point allows a direct proof that optimal solutions of the Fenchel-Rockafellar dual of the gauge dual are precisely the primal solutions rescaled by the optimal value. We extend the gauge duality framework to the setting in which the functional components are general nonnegative convex functions, including problems with piecewise linear quadratic functions and constraints that arise from generalized linear models used in regression.