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

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Abstract

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 in C\(++\) and the results of numerical experiments illustrate the efficacy of the proposed approach.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA, 18018, USA

    Frank E. Curtis & Xiaocun Que

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  1. Frank E. Curtis
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  2. Xiaocun Que
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Correspondence to Xiaocun Que.

Additional information

F. E. Curtis and X. Que were supported in part by National Science Foundation Grant DMS-1016291.

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Curtis, F.E., Que, X. A quasi-Newton algorithm for nonconvex, nonsmooth optimization with global convergence guarantees. Math. Prog. Comp. 7, 399–428 (2015). https://doi.org/10.1007/s12532-015-0086-2

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