论文标题
牛顿方法的最佳步长，在自我符合函数的最低限度附近
Optimal step length for the Newton method near the minimum of a self-concordant function
论文作者
论文摘要
在圆锥编程的路径遵循方法中，牛顿方法的性能在距离自我符合函数的最小化器有限距离的情况下，对于调整该方法的参数至关重要。有关进度和要使用的步长的可用界限是基于不同点的黑森人之间的保守关系，因此是优势。在此贡献中，我们使用最佳控制理论的方法来计算自我符合函数类别的牛顿方法的最佳步长，作为初始牛顿降低的函数，并导致最差的减少。确切的界限是根据无法明确集成的普通微分方程解决方案表示的。我们提供了近似的数值和分析表达式，这些表达式足以用于优化方法。作为一种应用，可以扩大需要保持路径跟随方法的迭代迭代术的中心路径的邻域，从而在每次迭代期间都能沿着中心路径更快地进展，从而更少迭代以达到给定的准确性。
In path-following methods for conic programming knowledge of the performance of the (damped) Newton method at finite distances from the minimizer of a self-concordant function is crucial for the tuning of the parameters of the method. The available bounds on the progress and the step length to be used are based on conservative relations between the Hessians at different points and are hence sub-optimal. In this contribution we use methods of optimal control theory to compute the optimal step length of the Newton method on the class of self-concordant functions, as a function of the initial Newton decrement, and the resulting worst-case decrease of the decrement. The exact bounds are expressed in terms of solutions of ordinary differential equations which cannot be integrated explicitly. We provide approximate numerical and analytic expressions which are accurate enough for use in optimization methods. As an application, the neighbourhood of the central path in which the iterates of path-following methods are required to stay can be enlarged, enabling faster progress along the central path during each iteration and hence fewer iterations to achieve a given accuracy.
