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On the use of third-order models with fourth-order regularization for unconstrained optimization
Optimization Letters ( IF 1.6 ) Pub Date : 2019-01-31 , DOI: 10.1007/s11590-019-01395-z E. G. Birgin , J. L. Gardenghi , J. M. Martínez , S. A. Santos
Optimization Letters ( IF 1.6 ) Pub Date : 2019-01-31 , DOI: 10.1007/s11590-019-01395-z E. G. Birgin , J. L. Gardenghi , J. M. Martínez , S. A. Santos
In a recent paper (Birgin et al. in Math Program 163(1):359–368, 2017), it was shown that, for the smooth unconstrained optimization problem, worst-case evaluation complexity \(O(\epsilon ^{-(p+1)/p})\) may be obtained by means of algorithms that employ sequential approximate minimizations of p-th order Taylor models plus \((p+1)\)-th order regularization terms. The aforementioned result, which assumes Lipschitz continuity of the p-th partial derivatives, generalizes the case \(p=2\), known since 2006, which has already motivated efficient implementations. The present paper addresses the issue of defining a reliable algorithm for the case \(p=3\). With that purpose, we propose a specific algorithm and we show numerical experiments.
中文翻译:
关于使用具有四阶正则化的三阶模型进行无约束优化
在最近的一篇论文中(Birgin等人在Math Program 163(1):359–368,2017)中显示,对于光滑无约束优化问题,最坏情况的评估复杂度\(O(\ epsilon ^ {- (p + 1)/ p})\)可以通过采用p阶泰勒模型和\((p + 1)\)阶正则化项的顺序近似最小化的算法来获得。假定p阶偏导数的Lipschitz连续性的上述结果推广了自2006年以来已知的情况\(p = 2 \),该情况已经激励了有效的实现。本文解决了为\(p = 3 \)情况定义可靠算法的问题。为此,我们提出了一种特定的算法,并进行了数值实验。
更新日期:2019-01-31
中文翻译:
关于使用具有四阶正则化的三阶模型进行无约束优化
在最近的一篇论文中(Birgin等人在Math Program 163(1):359–368,2017)中显示,对于光滑无约束优化问题,最坏情况的评估复杂度\(O(\ epsilon ^ {- (p + 1)/ p})\)可以通过采用p阶泰勒模型和\((p + 1)\)阶正则化项的顺序近似最小化的算法来获得。假定p阶偏导数的Lipschitz连续性的上述结果推广了自2006年以来已知的情况\(p = 2 \),该情况已经激励了有效的实现。本文解决了为\(p = 3 \)情况定义可靠算法的问题。为此,我们提出了一种特定的算法,并进行了数值实验。