Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijo-type line search

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. It is well-known that the direction generated by a conjugate gradient method may not be a descent direction of the objective function. In this paper, we take a little modification to the Fletcher–Reeves (FR) method such that the direction generated by the modified method provides a descent direction for the objective function. This property depends neither on the line search used, nor on the convexity of the objective function. Moreover, the modified method reduces to the standard FR method if line search is exact. Under mild conditions, we prove that the modified method with Armijo-type line search is globally convergent even if the objective function is nonconvex. We also present some numerical results to show the efficiency of the proposed method.Supported by the 973 project (2004CB719402) and the NSF foundation (10471036) of China.     

Global convergence of modified HS conjugate gradient method

In this paper, HS conjugate gradient method for minimizing a continuously differentiable functionf onR ⁿ is modified to have global convergence property. Firstly, it is shown that, using reverse modulus of continuity function and forcing function, the new method for solving unconstrained optimization can work for a continuously differentiable function with Curry-Altman’s step size rule and a bounded level set. Secondly, by using comparing technique, some general convergence properties of the new method with Armijo step size rule are established. Numerical results show that the new algorithms are efficient.     

The convergence of conjugate gradient method with nonmonotone line search

The conjugate gradient method is a useful and powerful approach for solving large-scale minimization problems. Liu and Storey developed a conjugate gradient method, which has good numerical performance but no global convergence under traditional line searches such as Armijo line search, Wolfe line search, and Goldstein line search. In this paper we propose a new nonmonotone line search for Liu-Storey conjugate gradient method (LS in short). The new nonmonotone line search can guarantee the global convergence of LS method and has a good numerical performance. By estimating the Lipschitz constant of the derivative of objective functions in the new nonmonotone line search, we can find an adequate step size and substantially decrease the number of functional evaluations at each iteration. Numerical results show that the new approach is effective in practical computation.     

推广线搜索下一类共轭梯度法的全局收敛性

在推广线搜索下给出了一类共轭梯度法的全局收敛结果     

Global convergence of some modified PRP nonlinear conjugate gradient methods

Recently, similar to Hager and Zhang (SIAM J Optim 16:170–192, 2005), Yu (Nonlinear self-scaling conjugate gradient methods for large-scale optimization problems. Thesis of Doctors Degree, Sun Yat-Sen University, 2007) and Yuan (Optim Lett 3:11–21, 2009) proposed modified PRP conjugate gradient methods which generate sufficient descent directions without any line searches. In order to obtain the global convergence of their algorithms, they need the assumption that the stepsize is bounded away from zero. In this paper, we take a little modification to these methods such that the modified methods retain sufficient descent property. Without requirement of the positive lower bound of the stepsize, we prove that the proposed methods are globally convergent. Some numerical results are also reported.     

Global convergence of a memory gradient method without line search

In this paper, we develop a memory gradient method for unconstrained optimization. The main characteristic of this method is that we obtain the next iteration without any line search. Under certain conditions, we obtain the strong global convergence of the proposed method.     

Global convergence of a two-parameter family of conjugate gradient methods without line search

We study the global convergence of a two-parameter family of conjugate gradient methods in which the line search procedure is replaced by a fixed formula of stepsize. This character is of significance if the line search is expensive in a particular application. In addition to the convergence results, we present computational results for various conjugate gradient methods without line search including those discussed by Sun and Zhang (Ann. Oper. Res. 103 (2001) 161–173).     

Global convergence of conjugate gradient method

In this article, we consider the global convergence of the Polak–Ribiére–Polyak conjugate gradient method (abbreviated PRP method) for minimizing functions that have Lipschitz continuous partial derivatives. A novel form of non-monotone line search is proposed to guarantee the global convergence of the PRP method. It is also shown that the PRP method has linear convergence rate under some mild conditions when the non-monotone line search reduces to a related monotone line search. The new non-monotone line search needs to estimate the Lipschitz constant of the gradients of objective functions, for which two practical estimations are proposed to help us to find a suitable initial step size for the PRP method. Numerical results show that the new line search approach is efficient in practical computation.     

带参数共轭梯度法簇的全局收敛性

共轭梯度法是最优化中最常用的方法之一,广泛地应用于求解大规模优化问题,其中参数β_k的不同选取可以构成不同的共轭梯度法.给出了一类含有三个参数的共轭梯度算法,这种算法能够在给定的条件下证明选定的β_k在每一步都能产生一个下降方向,同时在强Wolfe线搜索下,这种算法具有全局收敛性.