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Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time

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Abstract

The epsilon-subdifferential of convex univariate piecewise linear-quadratic (PLQ) functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using binary search, we improve the complexity to logarithmic worst-case time, and prove such complexity is optimal. In addition, a new algorithm to compute the entire graph of the epsilon-subdifferential in (optimal) linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined functions with little restrictions.

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Acknowledgements

This work was supported in part by Discovery Grant #298145-2013 (Lucet) from the Natural Sciences and Engineering Research Council of Canada (NSERC). Part of the research was performed in the Computer-Aided Convex Analysis (CA2) laboratory funded by a Leaders Opportunity Fund (LOF, John R. Evans Leaders Fund – Funding for research infrastructure) from the Canada Foundation for Innovation (CFI) and by a British Columbia Knowledge Development Fund (BCKDF).

This work was started at the end of Anuj Bajaj MSc research under the guidance of Dr. Warren Hare. Their preliminary efforts, ideas, and interest motivated the authors to pursue more efficient algorithms. The authors thank them for their initial contribution without which this work would not have been possible.

The authors are very grateful for several suggestions by the reviewers, which greatly improve the presentation of the paper.

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  1. Computer Science, UBC Okanagan, 3187 University Way, Kelowna, BC, V1V 1V7, Canada

    Deepak Kumar & Yves Lucet

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  1. Deepak Kumar
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  2. Yves Lucet
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Correspondence to Yves Lucet.

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Kumar, D., Lucet, Y. Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time. Set-Valued Var. Anal 27, 623–641 (2019). https://doi.org/10.1007/s11228-018-0476-5

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