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The Cost of Nonconvexity in Deterministic Nonsmooth Optimization

Author

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  • Siyu Kong

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853)

  • A. S. Lewis

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853)

Abstract
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ -stationary point of any directionally differentiable Lipschitz objective using O ( ϵ − 4 ) calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves O ( ϵ − 5 ) for any difference-of-convex objective and O ( ϵ − 4 ) for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.

Suggested Citation

  • Siyu Kong & A. S. Lewis, 2024. "The Cost of Nonconvexity in Deterministic Nonsmooth Optimization," Mathematics of Operations Research, INFORMS, vol. 49(4), pages 2385-2401, November.
    • Handle: RePEc:inm:ormoor:v:49:y:2024:i:4:p:2385-2401
    DOI: 10.1287/moor.2022.0289
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