To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definite incomplete Hessian matrix. It also shows that the Wolfe conditions hold in IHN with a line search step length of one. As an important application, an effective IHN and a modified IHN, called the truncated-IHN method (T-IHN), are constructed for solving a large scale chemical database optimal projection mapping problem. T-IHN is shown to work well even with indefinite incomplete Hessian matrices. Numerical results confirm the theoretical results of IHN, and demonstrate the promising potential of T-IHN as an efficient minimization algorithm.

中文翻译：

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一种不完全 Hessian Newton 最小化方法及其在化学数据库问题中的应用。

为了有效地解决具有稠密 Hessian 矩阵的大规模无约束最小化问题，本文提出使用不完全 Hessian 矩阵来定义一种新的改进牛顿方法，称为不完全 Hessian Newton 方法 (IHN)。理论分析表明，IHN全球会聚，并具有正确选择的对称，正定的粗糙矩阵的线性收敛速率。它还表明 Wolfe 条件在 IHN 中成立，线搜索步长为 1。作为一个重要的应用，构建了一个有效的 IHN 和一个改进的 IHN，称为截断 IHN 方法（T-IHN），用于解决大规模化学数据库的最优投影映射问题。T-IHN 被证明即使在不确定的不完全 Hessian 矩阵中也能很好地工作。数值结果证实了 IHN 的理论结果，