Implementing and modifying Broyden class updates for large scale optimization

Abstract

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.