Wu Yanlin
Department of Basic Teaching, Yango College, Fuzhou, Fujian 350015 P.R. China.
J Inequal Appl. 2017;2017(1):97. doi: 10.1186/s13660-017-1373-4. Epub 2017 May 3.
The nonlinear conjugate gradient (CG) algorithm is a very effective method for optimization, especially for large-scale problems, because of its low memory requirement and simplicity. Zhang (IMA J. Numer. Anal. 26:629-649, 2006) firstly propose a three-term CG algorithm based on the well known Polak-Ribière-Polyak (PRP) formula for unconstrained optimization, where their method has the sufficient descent property without any line search technique. They proved the global convergence of the Armijo line search but this fails for the Wolfe line search technique. Inspired by their method, we will make a further study and give a modified three-term PRP CG algorithm. The presented method possesses the following features: (1) The sufficient descent property also holds without any line search technique; (2) the trust region property of the search direction is automatically satisfied; (3) the steplengh is bounded from below; (4) the global convergence will be established under the Wolfe line search. Numerical results show that the new algorithm is more effective than that of the normal method.
非线性共轭梯度（CG）算法是一种非常有效的优化方法，特别是对于大规模问题，因为它内存需求低且简单。Zhang（《IMA数值分析杂志》26:629 - 649，2006）首先基于著名的无约束优化的Polak - Ribière - Polyak（PRP）公式提出了一种三项CG算法，其方法在没有任何线搜索技术的情况下具有充分下降性质。他们证明了Armijo线搜索的全局收敛性，但对于Wolfe线搜索技术则不成立。受他们方法的启发，我们将进行进一步研究并给出一种改进的三项PRP CG算法。所提出的方法具有以下特点：（1）在没有任何线搜索技术的情况下也具有充分下降性质；（2）搜索方向的信赖域性质自动满足；（3）步长有下界；（4）在Wolfe线搜索下将建立全局收敛性。数值结果表明新算法比常规方法更有效。
