Accelerated memory-less SR1 method with generalized secant equation for unconstrained optimization

The memory-less SR1 with generalized secant equation (MM-SR1gen) is presented and developed together with its numerical performances for solving a collection of 800 unconstrained optimization problems with the number of variables in the range [1000, 10000]. The convergence of the MM-SR1gen method is proved under the classical assumptions. Comparison between the MM-SR1gen versus the memory-less SR1 method, versus the memory-less BFGS method and versus the BFGS in implementation of Shanno and Phua from CONMIN show that MM-SR1gen is more efficient and more robust than these algorithms. By solving five applications from MINPACK-2 collection, each of them with 40,000 variables, we have the computational evidence that MM-SR1gen is more efficient than memory-less SR1 and than memory-less BFGS. The conclusion of this study is that the memory-less SR1 method with generalized secant equation is a rapid and reliable method for solving large-scale minimizing problems. Besides, it is shown that the accuracy of the Hessian approximations along the iterations in quasi-Newton methods is not as crucial in these methods as it is believed.