A Generalized Newton Method for Subgradient Systems
A Generalized Newton Method for Subgradient Systems
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DOI：
10.1287/moor.2022.1320
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发表时间：
2020-09
影响因子：
1.7
通讯作者：
Pham Duy Khanh;B. Mordukhovich;Vo Thanh Phat
中科院分区：
文献类型：
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作者：
Pham Duy Khanh;B. Mordukhovich;Vo Thanh Phat
本文提出并发展了一种新的Newton型算法来求解由广义实值正则函数的次梯度定义的次微分包含。该算法是制定在这样的功能，享有广泛的微积分规则，可以有效地计算广泛的类扩展实值函数的二阶次微分。在此基础上，利用次梯度映射的度量正则性和次正则性，建立了算法的适定性和局部超线性收敛性的可验证条件.所得到的结果对于由具有Lipschitz梯度的连续可微函数（[公式：见正文]函数）定义的方程类也是新的，这是我们考虑的基本情况。所开发的算法的正则函数和它们的扩展到一个结构类的复合函数制定的近似映射和前向后信封。除了众多的说明性的例子和比较与已知的算法[公式：见文字]功能和广义方程，本文提出的应用程序所提出的算法，正则化最小二乘问题中出现的统计，机器学习，和相关学科。资金来源：P.D.的研究。Khanh由胡志明市教育大学科学技术基金会资助[Grant CS.2022.19.20 TD]。B的研究。Mordukhovich和V.T. Phat部分由美国国家科学基金会资助[赠款DMS-1808978和DMS-2204519]。B. Mordukhovich还得到了空军科学研究办公室[Grant 15 RT 0462]和澳大利亚研究理事会在发现项目DP-190100555下的支持。这项工作得到了空军科学研究办公室的支持[Grant 15 RT 0462]。
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second order subdifferential of such functions that enjoy extensive calculus rules and can be efficiently computed for broad classes of extended real-valued functions. Based on this and on the metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring the well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian gradients ([Formula: see text] functions), which is the underlying case of our consideration. The developed algorithms for prox-regular functions and their extension to a structured class of composite functions are formulated in terms of proximal mappings and forward–backward envelopes. Besides numerous illustrative examples and comparison with known algorithms for [Formula: see text] functions and generalized equations, the paper presents applications of the proposed algorithms to regularized least square problems arising in statistics, machine learning, and related disciplines. Funding: Research of P. D. Khanh is funded by Ho Chi Minh City University of Education Foundation for Science and Technology [Grant CS.2022.19.20TD]. Research of B. Mordukhovich and V. T. Phat was partly supported by the U.S. National Science Foundation [Grants DMS-1808978 and DMS-2204519]. The research of B. Mordukhovich was also supported by the Air Force Office of Scientific Research [Grant 15RT0462] and the Australian Research Council under Discovery Project DP-190100555. This work was supported by the Air Force Office of Scientific Research [Grant 15RT0462].