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<p>The <i>cosine measure</i> was introduced in 2003 to quantify the richness of finite positive spanning sets of directions in the context of derivative-free directional methods. A positive spanning set is a set of vectors whose nonnegative linear combinations span the whole space. The present work extends the definition of cosine measure. In particular, the paper studies cosine measures relative to a subspace, and proposes a deterministic algorithm to compute it. The paper also studies the situation in which the set of vectors is infinite. The extended definition of the cosine measure might be useful for subspace decomposition methods.</p>

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The cosine measure relative to a subspace

  • 期刊论文

  • 发表日期：

    2025年6月6日

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  • Charles Audet,
  • Warren Hare,
  • Gabriel Jarry-Bolduc

摘要

The cosine measure was introduced in 2003 to quantify the richness of finite positive spanning sets of directions in the context of derivative-free directional methods. A positive spanning set is a set of vectors whose nonnegative linear combinations span the whole space. The present work extends the definition of cosine measure. In particular, the paper studies cosine measures relative to a subspace, and proposes a deterministic algorithm to compute it. The paper also studies the situation in which the set of vectors is infinite. The extended definition of the cosine measure might be useful for subspace decomposition methods.