Abstract
Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in \(\mathbb {R}^n\) has been established to be between \(n+1\) and 2n (with both extremes existing). The lower bound is immediate from being a positive spanning set, while the upper bound uses both positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in \(\mathbb {R}^n\) for \(n \in \{1, 2\}\) has at most 2n elements, but a positively linearly independent set in \(\mathbb {R}^n\) for \(n\ge 3\) can have an arbitrary number of elements.
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Acknowledgments
This research was partially funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Discover Grant #355571-2013, and by the Pacific Institute for the Mathematical Sciences (PIMS), “Optimization: Theory, Algorithms, and Applications” Collaborative Research Group. The authors are indebted to Dr. C. Audet, for helpful feedback and providing the new (shorter) proof to Proposition 3 which is presented in this paper.
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Hare, W., Song, H. On the cardinality of positively linearly independent sets. Optim Lett 10, 649–654 (2016). https://doi.org/10.1007/s11590-015-0959-3
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DOI: https://doi.org/10.1007/s11590-015-0959-3