The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

本文的主要重点是最小化Hessian为正定二次函数的精确线搜索方法。我们给出一类有限内存的拟牛顿黑森近似，其生成的搜索方向与BFGS方法的搜索方向平行，或者等效于与预处理共轭梯度方法的搜索方向相同。在简化的Hessian环境中，该类为构造有限内存的拟牛顿方法提供了动力框架。这些方法在精确算法中对二次优化问题实现了有限终止。通过对相关线性方程组的序列进行数值模拟，我们以有限精度算法显示了此框架内方法的性能，这些线性方程组源自CUTEst测试集合。此外，对于一般的无约束优化问题，我们在完整的Broyden类中给出了Hessian逼近的紧凑表示。此表示仅由显式矩阵和渐变组成，作为矢量分量。