Nonlinear conjugate gradient methods are among the most preferable and effortless methods to solve smooth optimization problems. Due to their clarity and low memory requirements, they are more desirable for solving large-scale smooth problems. Conjugate gradient methods make use of gradient and the previous direction information to determine the next search direction, and they require no numerical linear algebra. However, the utility of nonlinear conjugate gradient methods has not been widely employed in solving nonsmooth optimization problems. In this paper, a modified nonlinear conjugate gradient method, which achieves the global convergence property and numerical efficiency, is proposed to solve large-scale nonsmooth convex problems. The new method owns the search direction, which generates sufficient descent property and belongs to a trust region. Under some suitable conditions, the global convergence of the proposed algorithm is analyzed for nonsmooth convex problems. The numerical efficiency of the proposed algorithm is tested and compared with some existing methods on some large-scale nonsmooth academic test problems. The numerical results show that the new algorithm has a very good performance in solving large-scale nonsmooth problems.

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一种用于大规模非光滑凸优化的修正非线性共轭梯度算法

非线性共轭梯度方法是解决平滑优化问题的最优选和最轻松的方法之一。由于它们的清晰度和低内存要求，它们更适合解决大规模平滑问题。共轭梯度方法利用梯度和先前的方向信息来确定下一个搜索方向，并且不需要数值线性代数。然而，非线性共轭梯度方法的实用性尚未广泛用于解决非光滑优化问题。本文提出了一种具有全局收敛性和数值效率的改进非线性共轭梯度法来求解大规模非光滑凸问题。新方法拥有搜索方向，产生足够的下降属性并属于一个信任区域。在一些合适的条件下，分析了该算法对非光滑凸问题的全局收敛性。在一些大规模的非光滑学术测试问题上，对所提算法的数值效率进行了测试，并与现有的一些方法进行了比较。数值结果表明，新算法在求解大规模非光滑问题时具有很好的性能。