Mathematical Programming Computation, Volume 7, Issue 4, December 2015

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A quasi-Newton algorithm for nonconvex, nonsmooth optimization with global convergence guarantees

Xiaocun Que, Frank E. Curtis


A line search algorithm for minimizing nonconvex and/or nonsmooth objective functions is presented. The algorithm is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) and an adaptive gradient sampling (GS) method. The BFGS strategy is employed because it typically yields fast convergence to the vicinity of a stationary point, and together with the adaptive GS strategy the algorithm ensures that convergence will continue to such a point. Under suitable assumptions, it is proved that the algorithm converges globally with probability one. The algorithm has been implemented inC++and the results of numerical experiments illustrate the efficacy of the proposed approach.

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