We consider Broyden class updates for large scale optimization problems in n dimensions, restricting attention to the case when the initial second derivative approximation is the identity matrix. Under this assumption we present an implementation of the Broyden class based on a coordinate transformation on each iteration. It requires only \(2nk + O(k^{2}) + O(n)\) multiplications on the kth iteration and stores \(nK+ O(K^2) + O(n)\) numbers, where K is the total number of iterations. We investigate a modification of this algorithm by a scaling approach and show a substantial improvement in performance over the BFGS method. We also study several adaptations of the new implementation to the limited memory situation, presenting algorithms that work with a fixed amount of storage independent of the number of iterations. We show that one such algorithm retains the property of quadratic termination. The practical performance of the new methods is compared with the performance of Nocedal’s (Math Comput 35:773--782, 1980) method, which is considered the benchmark in limited memory algorithms. The tests show that the new algorithms can be significantly more efficient than Nocedal’s method. Finally, we show how a scaling technique can significantly improve both Nocedal’s method and the new generalized conjugate gradient algorithm.

我们考虑针对n维中的大规模优化问题的Broyden类更新，从而将注意力集中在初始二阶导数近似为单位矩阵的情况下。在此假设下，我们基于每次迭代的坐标变换展示了Broyden类的实现。它在第k次迭代中只需要\（2nk + O（k ^ {2}）+ O（n）\）乘法，并存储\（nK + O（K ^ 2）+ O（n）\）个数字，其中K是迭代的总数。我们研究了通过缩放方法对该算法的修改，并显示了与BFGS方法相比性能的显着提高。我们还研究了新实现对有限内存情况的几种改编，并提出了与固定数量的存储无关的算法，这些算法与迭代次数无关。我们证明了一种这样的算法保留了二次终止的性质。将新方法的实际性能与Nocedal（Math Comput 35：773--782，1980）方法的性能进行比较，后者被认为是有限内存算法中的基准。测试表明，新算法比Nocedal的方法效率更高。最后，