用于无约束极小化的动态邻近点方法
Dynamic Proximal Point Method for Unconstrained Minimization
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中文总结 AI 辅助
针对无约束极小化问题，提出自适应调整对角正则矩阵的动态邻近点算法，结合内层牛顿法与线搜索保证收敛，给出实现伪代码与停止准则，为无约束优化提供新方法。
中文摘要 AI 辅助
在本研究中，我们提出一种新型的动态邻近点算法，用于无约束优化。该方法生成一系列邻近子问题，其中二次正则项由对角矩阵加权，该对角矩阵在每次迭代中自适应更新。每个子问题采用结合线搜索的内层牛顿法求解，为非线性求解器提供全局收敛机制。在外层，算法根据内层牛顿求解器的性能更新参考点并调整正则化参数。我们推导用于计算牛顿步的简化线性系统，定义相应的 merit 函数（价值函数），并讨论基于导数信息构造对角缩放矩阵的实用方法。本文还提供面向实现的伪代码及与所提方法一致的停止准则。
英文摘要
In this work, we present a novel dynamic proximal point algorithm for unconstrained optimization. The method generates a sequence of proximal subproblems, where the quadratic regularization term is weighted by a diagonal matrix that is updated adaptively at each iteration. Each subproblem is solved using an inner Newton's method combined with a line search, which provides a global convergence mechanism for the nonlinear solver. At the outer level, the algorithm updates the reference point and adjusts the regularization parameter based on the performance of the inner Newton solver. We derive the reduced linear system used to compute the Newton step, define the corresponding merit function, and discuss practical approaches for constructing the diagonal scaling matrix from derivative information. The paper also provides implementation-oriented pseudocode and stopping criteria that are consistent with the proposed method.
