In this paper, a new subspace minimization conjugate gradient method based on modified secant equation is proposed and analyzed. For a classical subspace minimization conjugate gradient method, the search direction is derived by minimizing an approximate quadratic model of objective function in a two-dimensional subspace. Generally, the approximate Hessian matrix in the above quadratic model is required to satisfy the standard secant equation, while we consider an approximate Hessian matrix which satisfies the modified secant equation. We give some rules such that if these rules are satisfied, we choose the standard secant equation, otherwise we choose the modified one. We can prove that the proposed directions satisfy the sufficient descent property under some extra conditions. We also present a modified nonmonotone Wolfe line search and establish the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with famous CG_DESCENT (5.3) (Hager and Zhang in SIAM J Optim 16(1):170–192, 2005) and SMCG_BB (Liu and Liu in J Optim Theory Appl 180(3):879–906, 2019), and show that the proposed algorithm is very promising.

中文翻译：

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基于修正割线方程的子空间最小化共轭梯度法无约束优化

提出并分析了一种基于修正割线方程的子空间最小共轭梯度法。对于经典子空间最小化共轭梯度法，搜索方向是通过最小化二维子空间中目标函数的近似二次模型得出的。通常，需要上述二次模型中的近似Hessian矩阵来满足标准割线方程，而我们考虑满足修正割线方程的近似Hessian矩阵。我们给出一些规则，如果满足这些规则，则选择标准割线方程，否则我们选择修改后的方程。我们可以证明所提出的方向在某些额外条件下满足足够的下降特性。我们还提出了一种改进的非单调Wolfe线搜索，并在温和的假设下建立了针对一般非线性函数的拟议方法的全局收敛性。使用著名的CG_DESCENT（5.3）（Hager和Zhang在SIAM J Optim 16（1）：170–192，2005）和SMCG_BB（Liu和Liu在J Optim Theory Appl 180（3）：879–906，2019）中进行了数值比较），并且表明所提出的算法非常有前途。