HOW UP-TO-DATE ARE LOW-RANK UPDATES?

Abstract

For several decades quasi-Newton methods based on low-rank secant updates have been widely applied to many small to medium sized nonlinear equations and optimization problems. Their adaptation to large and structured problems has not always been successful. We review some convergence results for secant methods and some examples regarding the cost of derivative matrices, report some recent results from a parallel implementation of Broyden's method, and propose an unsymmetric rank-one Jacobian update based on direct and adjoint derivative information. It may be applied in particular to Jacobians in constrained optimization, either with full storage or in a limited memory version. We report some numerical results on a discretized second order ODE and conclude with an outlook on future developments