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Reformulation of the M-Stationarity Conditions as a System of Discontinuous Equations and Its Solution by a Semismooth Newton Method
SIAM Journal on Optimization ( IF 3.1 ) Pub Date : 2021-05-27 , DOI: 10.1137/20m1321413
Felix Harder , Patrick Mehlitz , Gerd Wachsmuth

SIAM Journal on Optimization, Volume 31, Issue 2, Page 1459-1488, January 2021.
We show that the Mordukhovich-stationarity system associated with a mathematical program with complementarity constraints (MPCC) can be equivalently written as a system of discontinuous equations which can be tackled with a semismooth Newton method. It will be demonstrated that the resulting algorithm can be interpreted as an active set strategy for MPCCs. Local fast convergence of the method is guaranteed under validity of an MPCC-tailored version of LICQ and a suitable strong second-order condition. In case of linear-quadratic MPCCs, the LICQ-type constraint qualification can be replaced by a weaker condition which depends on the underlying multipliers. We discuss a suitable globalization strategy for our method. Some numerical results are presented in order to illustrate our theoretical findings.


中文翻译:

用半光滑牛顿法将 M 平稳条件改写为不连续方程组及其解

SIAM Journal on Optimization,第 31 卷,第 2 期,第 1459-1488 页,2021 年 1 月。
我们表明,与具有互补约束 (MPCC) 的数学程序相关联的 Mordukhovich 平稳系统可以等效地写成一个不连续方程组,可以用半光滑牛顿方法处理。将证明由此产生的算法可以解释为 MPCC 的活动集策略。该方法的局部快速收敛在 LICQ 的 MPCC 定制版本的有效性和合适的强二阶条件下得到保证。在线性-二次型 MPCC 的情况下,LICQ 类型的约束条件可以用一个较弱的条件代替,该条件取决于基础乘法器。我们为我们的方法讨论了一个合适的全球化策略。为了说明我们的理论发现,给出了一些数值结果。
更新日期:2021-06-03
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