A special complementarity function revisited
A special complementarity function revisited
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DOI：
10.1080/02331934.2018.1470177
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发表时间：
2019-01
期刊：
影响因子：
2.2
通讯作者：
R. Behling;A. Fischer;K. Schönefeld;Nico Strasdat
中科院分区：
文献类型：
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作者：
R. Behling;A. Fischer;K. Schönefeld;Nico Strasdat
摘要最近，建立了约束方程组牛顿型方法的局部框架。应用于Karush-Kuhn-Tucker （KKT）系统的解，该框架在允许非孤立和退化KKT点的条件下实现了局部二次收敛。这个结果是基于KKT条件作为一个约束分段光滑方程组的重新表述。对于其他（不是分段平滑的）重新配方是否能取得类似的结果，这是一个悬而未决的问题。如果在允许退化KKT点和非孤立拉格朗日乘子的条件下，用fisher - burmeister互补函数重新表述KKT系统，这是可能的。为此，引入了一种新的约束Levenberg-Marquardt子问题。它们允许更长的步骤来更新乘数。在此基础上，收敛速度至少为1.5。
ABSTRACT Recently, a local framework of Newton-type methods for constrained systems of equations has been developed. Applied to the solution of Karush–Kuhn–Tucker (KKT) systems, the framework enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of equations. It is an open question whether a comparable result can be achieved for other (not piecewise smooth) reformulations. It will be shown that this is possible if the KKT system is reformulated by means of the Fischer–Burmeister complementarity function under conditions that allow degenerate KKT points and nonisolated Lagrange multipliers. To this end, novel constrained Levenberg–Marquardt subproblems are introduced. They allow significantly longer steps for updating the multipliers. Based on this, a convergence rate of at least 1.5 is shown.