In this paper, we suggest a novel approach termed as regularized based implicit Lagrangian twin extreme learning machine in primal as a pair of unconstrained convex minimization problem (RILTELM) where regularization term is added to follow the structural risk minimization principle. Here, we consider 2-norm of the slack vector of variables to make the problem strongly convex which results in a unique solution. Since it has non-smooth plus functions in their objective function, so we find an approximate solution by replacing the non-smooth plus function with smooth approximation function because to find an approximation solution in primal space is always superior to its dual. Due to non-smooth plus function, we solve the problem by either smooth approximation approach or generalized derivative approach. In addition, a functional iterative scheme is also suggested to find the optimal solution. Hence, no external optimization toolbox is required unlike in twin extreme learning machine (TELM) and twin support vector machine (TWSVM). The numerical experiments are demonstrated on artificial and real-world datasets and compared with TWSVM, ELM, TELM and LSTELM to establish the efficacy and applicability of proposed RILTELM.

在本文中，我们提出了一种新颖的方法，称为原始的基于正则化的隐式拉格朗日双极端学习机，作为一对无约束凸最小化问题（RILTELM），其中添加了正则化项以遵循结构风险最小化原则。在这里，我们考虑变量的松弛向量的2范数，以使问题强凸，从而得出唯一的解。由于它在目标函数中具有非光滑加函数，因此我们用光滑逼近函数代替非光滑加函数来找到近似解，因为在原始空间中找到逼近解总是优于其对偶。由于不光滑的加函数，我们通过光滑逼近法或广义导数法来解决问题。此外，还建议使用功能迭代方案来找到最佳解决方案。因此，与双胞胎极限学习机（TELM）和双胞胎支持向量机（TWSVM）不同，不需要外部优化工具箱。在人工和真实的数据集上进行了数值实验，并与TWSVM，ELM，TELM和LSTELM进行了比较，以建立所提出的RILTELM的有效性和适用性。