Abstract
A simplex, the convex hull of a set of \(n+1\) affinely independent points, is a useful tool in derivative-free optimization. The term uniform simplex was used by Audet and Hare (Derivative-free and blackbox optimization. Springer series in operations research and financial engineering, Springer, Cham, 2017). The purpose of this paper is to provide a framework for constructing a uniform simplex in \(\mathbb {R}^n\), which can then be aligned with a given vector. We prove that a uniform simplex has the greatest normalized volume of any simplex. Moreover, we show how to create a uniform minimal positive basis from a uniform simplex.
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References
Abramson, M.A., Audet, C., Dennis Jr., J.E., Le Digabel, S.: OrthoMADS: a deterministic MADS instance with orthogonal directions. SIAM J. Optim. 20, 948–966 (2009)
Alberto, P., Nogueira, F., Rocha, H., Vicente, L.: Pattern search methods for user-provided points: application to molecular geometry problems. SIAM J. Optim. 14, 1216–1236 (2004)
Audet, C.: A short proof on the cardinality of maximal positive bases. Optim. Lett. 5, 191–194 (2011)
Audet, C., Dennis Jr., J.E.: Analysis of generalized pattern searches. SIAM J. Optim. 13(2002), 889–903 (2003)
Audet, C., Hare, W.: Derivative-Free and Blackbox Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Cham (2017)
Audet, C., Ianni, A., Le Digabel, S., Tribes, C.: Reducing the number of function evaluations in mesh adaptive direct search algorithms. SIAM J. Optim. 24, 621–642 (2014)
Conn, A., Scheinberg, K., Vicente, L.: Introduction to Derivative-Free Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Coope, I.D., Price, C.J.: Positive bases in numerical optimization. Comput. Optim. Appl. 21, 169–175 (2002)
Davis, C.: Theory of positive linear dependence. Am. J. Math. 76, 733 (1954)
Golan, J.: The Linear Algebra a Beginning Graduate Student Ought to Know. Springer, Berlin (2012)
Lewis, R.M., Torczon, V.: Rank ordering and positive bases in pattern search algorithms. Technology Reports. Institute for computer applications in science and engineering VA (1996)
Loehr, N.: Advanced Linear Algebra. Chapman and Hall, Boca Raton (2014)
McKinnon, K.: Convergence of the Nelder–Mead simplex method to a nonstationary point. SIAM J. Optim. 9, 148–158 (1998)
Nelder, J., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)
Regis, R.: On the properties of positive spanning sets and positive bases. Optim. Eng. 17, 229–262 (2016)
Torczon, V.: On the convergence of pattern search algorithms. SIAM J. Optim. 7, 1–25 (1997)
Acknowledgements
We would like to acknowledge Dr. Warren Hare for his valuable and constructive suggestions. 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). Jarry-Bolduc’s research is partially funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada, Discover Grant #2018-03865. Nadeau’s research is partially funded by NSERC of Canada, Discover Grant #2018-03865. Singh’s research is partially funded by NSERC of Canada, Discover Grant #298145-2013 and #2018-03928 (Lucet), and MITACS.
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Jarry-Bolduc, G., Nadeau, P. & Singh, S. Uniform simplex of an arbitrary orientation. Optim Lett 14, 1407–1417 (2020). https://doi.org/10.1007/s11590-019-01448-3
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DOI: https://doi.org/10.1007/s11590-019-01448-3