The $M_2$ variables are devised to extend $M_{T2}$ by promoting transverse masses to Lorentz-invariant ones and making explicit use of on-shell mass relations. Unlike simple kinematic variables such as the invariant mass of visible particles, where the variable definitions directly provide how to calculate them, the calculation of the $M_2$ variables is undertaken by employing numerical algorithms. Essentially, the calculation of $M_2$ corresponds to solving a constrained minimization problem in mathematical optimization, and various numerical methods exist for the task. We find that the sequential quadratic programming method performs very well for the calculation of $M_2$, and its numerical performance is even better than the method implemented in the existing software package for $M_2$. As a consequence of our study, we have developed and released yet another software library, YAM2, for calculating the $M_2$ variables using several numerical algorithms.

Program summary

Program title: YAM2

CPC Library link to program files: https://doi.org/10.17632/4g7wfd5fpb.1

Developer’s repository link: https://github.com/cbpark/YAM2

Licensing provisions: BSD 3-Clause

Programming language: C ++

Nature of problem: The value and the solution of the $M_2$ variables can be obtained from the optimality and feasibility conditions of the nonlinearly constrained minimization problem in numerical optimization. To perform the calculation properly, one should employ suitable numerical algorithms with the appropriate formulation of the variables, having in mind the algorithmic efficiency and the computational cost.

Solution method: There exist various numerical methods for solving constrained optimization problems. We have chosen the sequential quadratic programming method with the derivative-dependent quasi-Newton algorithm since it performs very efficiently to find the local minimum using derivative information. The method has been codified by using the implementation of the numerical algorithms in the NLopt library [1]. The new library also includes the routines using other algorithms for calculating $M_2$, such as the augmented Lagrangian method.

References:

S. G. Johnson, “The NLopt nonlinear-optimization package,” https://github.com/stevengj/nlopt

中文翻译：

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YAM2：另一个图书馆 ${\mathrm{中号}}_{\mathrm{2个}}$ 使用顺序二次编程的变量

这 ${\mathrm{中号}}_{\mathrm{2个}}$ 设计变量以扩展 ${\mathrm{中号}}_{Ť\mathrm{2个}}$通过将横向质量提升到洛仑兹不变的质量，并明确使用壳上质量关系。与简单的运动学变量（例如可见粒子的不变质量）不同，在变量定义中直接提供了如何计算它们的方法，${\mathrm{中号}}_{\mathrm{2个}}$变量是通过采用数值算法进行的。从本质上讲，${\mathrm{中号}}_{\mathrm{2个}}$对应于在数学优化中解决约束最小化问题，并且为此任务存在各种数值方法。我们发现，顺序二次规划方法在计算${\mathrm{中号}}_{\mathrm{2个}}$，其数值性能甚至优于现有软件包中实现的方法 ${\mathrm{中号}}_{\mathrm{2个}}$。这项研究的结果是，我们开发并发布了另一个软件库YAM2，用于计算${\mathrm{中号}}_{\mathrm{2个}}$ 使用几种数值算法的变量。

计划摘要

节目名称： YAM2

CPC库链接到程序文件： https : //doi.org/10.17632/4g7wfd5fpb.1

开发人员的资料库链接： https : //github.com/cbpark/YAM2

许可条款： BSD 3条款

编程语言： C ++

问题的性质：解决方案的价值和解决方案${\mathrm{中号}}_{\mathrm{2个}}$可以从数值优化中的非线性约束最小化问题的最优性和可行性条件中获得变量。为了正确执行计算，应考虑算法效率和计算成本，并使用具有适当变量表述的合适数值算法。

求解方法：存在多种用于求解约束优化问题的数值方法。我们选择了依赖于导数的拟牛顿算法的顺序二次编程方法，因为它可以非常有效地利用导数信息找到局部最小值。该方法已通过使用NLopt库中的数值算法实现进行了编码[1]。新的库还包括使用其他算法进行计算的例程${\mathrm{中号}}_{\mathrm{2个}}$，例如增强型拉格朗日方法。

参考：

SG Johnson，“ NLopt非线性优化程序包”，https：//github.com/stevengj/nlopt