Abstract

New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the Hessian matrix is singular at the solution point of the function to be minimized, the gradient of the objective function belongs to the image of its second derivative. Due to this new property of convex functions, Newtonian methods for solving unconstrained optimization problems can be applied without assuming the nonsingularity of the Hessian matrix at the solution of the problem and their rate of convergence in argument can be estimated under fairly general assumptions.

中文翻译：

---

光滑凸函数的一些性质和牛顿法

摘要

建立了与极值问题相关的凸无限微分函数的新性质。结果表明，在解的邻域内，即使Hessian矩阵在待最小化函数的解点处是奇异的，目标函数的梯度也属于其二阶导数的图像。由于凸函数的这一新特性，可以应用牛顿方法来解决无约束优化问题，而无需假设解决问题时 Hessian 矩阵的非奇异性，并且可以在相当一般的假设下估计它们在参数上的收敛速度。