最优化是应用数学的一个分支,主要指在一定条件限制下,选取某种研究方案使目标达到最优的一种方法。最优化问题在当今的军事、工程、管理等领域有着极其广泛的应用。

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我们提出并分析了一种基于动量的梯度方法,用于训练具有指数尾损失(例如,指数或logistic损失)的线性分类器,它以O (1/t2)的速率最大化可分离数据的分类边缘。这与标准梯度下降的速率O(1/log(t))和标准化梯度下降的速率O(1/t)形成对比。这种基于动量的方法是通过最大边际问题的凸对偶,特别是通过将Nesterov加速度应用于这种对偶,从而在原函数中得到了一种简单而直观的方法。这种对偶观点也可以用来推导随机变量,通过对偶变量进行自适应非均匀抽样。

https://www.zhuanzhi.ai/paper/9fd848dc95d2b0a9a5da37dbbd79d4ed

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The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained.

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