Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Larson, Jeffrey; Menickelly, Matt; Zhou, Baoyu

doi:10.1137/20m1378089

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Journal Article · Thu Oct 28 00:00:00 EDT 2021 · SIAM Journal on Optimization

DOI:https://doi.org/10.1137/20m1378089· OSTI ID:1839939

^[1]; ^[1]; Zhou, Baoyu ^[2]

Argonne National Lab. (ANL), Argonne, IL (United States)
Lehigh Univ., Bethlehem, PA (United States)

Here we propose a manifold sampling algorithm for minimizing a nonsmooth composition $$f= h\circ F$$, where we assume $$h$$ is nonsmooth and may be inexpensively computed in closed form and $$F$$ is smooth but its Jacobian may not be available. We additionally assume that the composition $$h\circ F$$ defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of $$F$$ are combined with particularly sampled information about $$h$$ to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about $$f$$.

View Accepted Manuscript (DOE)

Research Organization:: Argonne National Laboratory (ANL), Argonne, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)

Grant/Contract Number:: AC02-06CH11357

OSTI ID:: 1839939

Journal Information:: SIAM Journal on Optimization, Journal Name: SIAM Journal on Optimization Journal Issue: 4 Vol. 31; ISSN 1052-6234

Publisher:: Society for Industrial and Applied Mathematics (SIAM)Copyright Statement

Country of Publication:: United States

Language:: English

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