AI 中文总结
研究针对集优化问题，引入非线性Hager-Zhang共轭梯度法，通过Drummond-Svaiter标量化函数讨论Wolfe线搜索条件，引入共轭参数得出搜索方向，证明其为下降方向，建立渐近全局收敛性，经数值实验验证了该方法的性能和有效性。
AI 中文摘要
本文介绍了一种用于解决集优化问题的非线性Hager-Zhang共轭梯度法。所考虑的目标函数由有限个连续可微函数定义。该方法对序锥有限生成元的存在性以及最优解处的正则性条件均无限制，对集优化和向量优化问题都有重要意义，向量优化是集优化的特殊情况。研究先讨论使用Drummond-Svaiter标量化函数的Wolfe线搜索条件，确定满足条件的步长存在性，引入Hager-Zhang标量共轭参数得出搜索方向，证明其为下降方向并给出方法的定义。还讨论重要结果和类Zoutendijk条件以确保全局收敛，最终建立了该方法的渐近全局收敛性。通过数值实验验证了所提技术的实际性能和有效性。
英文摘要
This work introduces a nonlinear Hager-Zhang conjugate gradient method for solving set optimization problems. The objective function under consideration is defined by a finite collection of continuously differentiable functions. Notably, the proposed approach imposes restrictions neither on the existence of a finite generator of the ordering cone nor on any regularity condition at the optimal solution. As a result, the proposed method holds considerable significance for both set optimization and vector optimization problems, with the latter serving as a special case of the former. The study begins by discussing Wolfe line search conditions using Drummond-Svaiter scalarization function. Thereafter, we establish the existence of a step length satisfying the Wolfe line search conditions along a descent direction. The Hager-Zhang scalar conjugate parameter is introduced to derive the search direction for the proposed method. It is established that the direction generated by the proposed method is a descent direction. The well-definedness of the proposed method is given. Furthermore, we discuss some important results and a Zoutendijk-like condition to ensure global convergence. Subsequently, the global convergence of the proposed method is established in an asymptotic manner. Finally, numerical experiments on various test problems validate the practical performance and effectiveness of the proposed technique.
