### Abstract

The performance of some modern optimization techniques in fitting the solutions of first-order differential equations derived from compartmented models to isotopic tracer data has been investigated. The methods studied were the conjugate gradient methods of Fletcher and Powell and Fletcher and Reeves, and the simplex method of Nelder and Mead. Fletcher and Powell's method was the least satisfactory of the three. When started on a simple problem outside the quadratic region, it approached the solution tangentially in a very inefficient way. Fletcher and Reeves' method solved this problem efficiently, but failed to find the true optimum in a problem of moderate complexity. Neither of the conjugate gradient methods would converge unless the parameters were scaled internally. The simplex method created its own scaling factors and worked more satisfactorily than either of the others on the simple and moderate problems. On a large problem (eight parameters) the performances of the simplex method without scaling and Fletcher and Reeves' method with scaling were about equal. The former was thus the most satisfactory of the three studied. With one difficult set of data, all methods found local minima whose coordinates depended on the starting point. The possibility of locating multiple minima cannot therefore be dismissed out of hand. Normalization is important when tracer data are involved, because they can vary over several orders of magnitude. When replicate data are involved, the variance at each time point may be used. Other possibilities are recommended when only one set of data is available. © 1972.

Original language | English |
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Pages (from-to) | 265-282 |

Number of pages | 18 |

Journal | Mathematical Biosciences |

Volume | 13 |

Issue number | 3-4 |

Publication status | Published - Apr 1972 |

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## Cite this

*Mathematical Biosciences*,

*13*(3-4), 265-282.