This paper proposes an optimization approach to find a set of orthogonal intrinsic mode functions (IMFs). In particular, an optimization problem is formulated in such a way that the total energy of the difference between the original IMFs and the corresponding obtained IMFs is minimized subject to both the orthogonal condition and the IMF conditions. This formulated optimization problem consists of an exclusive or constraint. This exclusive or constraint is further reformulated to an inequality constraint. Using the Lagrange multiplier approach, it is required to solve a linear matrix equation, a quadratic matrix equation and a highly nonlinear matrix equation only dependent on the orthogonal IMFs as well as a nonlinear matrix equation dependent on both the orthogonal IMFs and the Lagrange multipliers. To solve these matrix equations, the first three equations are considered. First, a new optimization problem is formulated in such a way that the error energy of the highly nonlinear matrix equation is minimized subject to the linear matrix equation and the quadratic matrix equation. By finding the nearly global optimal solution of this newly formulated optimization problem and checking whether the objective functional value evaluated at the obtained solution is close to zero or not, the orthogonal IMFs are found. Finally, by substituting the obtained orthogonal IMFs to the last matrix equation, this last matrix equation reduced to a linear matrix equation which is only dependent on the Lagrange multipliers. Therefore, the Lagrange multipliers can be found. Consequently, the solution of the original optimization problem is found. By repeating these procedures with different initial conditions, a nearly global optimal solution is obtained.

中文翻译：

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正交本征函数的优化方法

本文提出了一种优化方法来查找一组正交本征模式函数（IMF）。特别地，以这样的方式来制定优化问题，即，在正交条件和IMF条件下，原始IMF和相应获得的IMF之间的差的总能量被最小化。这个公式化的优化问题由排他或约束组成。将该排他或约束进一步重构为不平等约束。使用拉格朗日乘数法，需要求解仅依赖于正交IMF的线性矩阵方程，二次矩阵方程和高度非线性矩阵方程，以及依赖于正交IMF和拉格朗日乘数的非线性矩阵方程。为了解决这些矩阵方程，考虑前三个方程。首先，以线性矩阵方程和二次矩阵方程最小化高度非线性矩阵方程的误差能量的方式提出新的优化问题。通过找到这个新提出的优化问题的几乎全局最优解，并检查在所获得的解中评估的目标函数值是否接近零，可以找到正交IMF。最后，通过将获得的正交IMF代入最后一个矩阵方程，可以将该最后一个矩阵方程简化为仅依赖于Lagrange乘数的线性矩阵方程。因此，可以找到拉格朗日乘数。因此，找到了原始优化问题的解决方案。