The recently developed Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems relies on solving a functional minimization problem for the most probable path, which is then utilized for evaluating a specific point of the system joint response probability density function (PDF). However, although various numerical optimization algorithms can be employed for determining the WPI most probable path, there is generally no guarantee that the selected algorithm converges to a global extremum.

In this paper, first, a Newton’s optimization scheme is proposed for determining the most probable path, and various convergence behavior aspects are elucidated. Second, the existence of a unique global minimum and the convexity of the objective function of the considered nonlinear system are demonstrated by resorting to computational algebraic geometry concepts and tools, such as Gröbner bases. Several numerical examples pertaining to diverse nonlinear oscillators are considered, where it is proved that the associated objective functions are convex, and that the proposed Newton’s scheme converges to the globally optimum most probable path. Comparisons with pertinent Monte Carlo simulation data are included as well for demonstrating the reliability of the WPI technique.

最近开发的用于确定各种非线性系统随机响应的Wiener路径积分（WPI）技术依赖于解决最可能路径的函数最小化问题，然后将其用于评估系统联合响应概率密度函数的特定点（ PDF）。但是，尽管可以使用各种数值优化算法来确定WPI最可能的路径，但是通常不能保证所选算法收敛于全局极值。

在本文中，首先，提出了牛顿优化方案来确定最可能的路径，并阐明了各种收敛行为方面。其次，通过使用计算代数几何概念和工具，例如Gröbner基，证明了所考虑的非线性系统存在唯一全局最小值和目标函数的凸性。考虑了与各种非线性振荡器有关的几个数值示例，其中证明了相关的目标函数是凸的，并且所提出的牛顿法收敛于全局最优的最可能路径。还包括与相关蒙特卡洛模拟数据的比较，以证明WPI技术的可靠性。