Abstract
The conical algorithm is a global optimization algorithm proposed by Tuy in 1964 to solve concave minimization problems. Introducing the concept of pseudo-nonsingularity, we give an alternative proof of convergence of the algorithm with the \(\omega \)-subdivision rule. We also develop a new convergent subdivision rule, named \(\omega \)-bisection, and report numerical results of comparing it with the usual \(\omega \)-subdivision.
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Notes
In some literature, the term “\(\gamma \)-valid cut” refers to the closure of the complement of \(G\).
Instead of “nonsingular”, Tuy used the term “nondegenerate” derived from an analogous concept in [4]. However, since it is easily confused with nondegeneracy in linear programming, we use “nonsingular” in view of its relation to the invertibility of \({\mathbf{Q}}_k\). Also the definition here follows that in [5].
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The authors would like to thank the anonymous referee for his/her valuable comments, which significantly improved the quality of this article.
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Takahito Kuno was partially supported by a Grant-in-Aid for Scientific Research (C 25330022) from Japan Society for the Promotion of Science.
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Kuno, T., Ishihama, T. A convergent conical algorithm with \(\omega \)-bisection for concave minimization. J Glob Optim 61, 203–220 (2015). https://doi.org/10.1007/s10898-014-0197-8
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DOI: https://doi.org/10.1007/s10898-014-0197-8