# PharmPK Discussion - Robustness of the Simplex method

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• On 24 Aug 1998 at 11:26:14, "Craig Hendrix. M.D." (chendrix.-at-.erols.com) sent the message
`I was cleaning out some old messages and looked at this one again.  Canyou point to some literature that discusses the robustness of theSimplex algorithm?ThanksCraig HendrixJohns Hopkins`
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• On 25 Aug 1998 at 10:45:12, David_Bourne (david.-a-.pharm.cpb.uokhsc.edu) sent the message
`[Two replies - db]X-Sender: st005899.-a-.brandywine.otago.ac.nzMime-Version: 1.0Date: Tue, 25 Aug 1998 11:32:20 +1200To: PharmPK.aaa.pharm.cpb.uokhsc.eduFrom: Robert Purves Subject: Re: PharmPK Robustness of the Simplex methodI do not know of any published comparisons. The following extracts from theinstruction manual for Minim are relevant. (Minim is a non-linear leastsquares program for the Macintosh family of computers). The name "Simplex"is apt to be misleading, since there is an unrelated numerical programmingtechnique with that name, and I prefer "Nelder-Mead" or "polytope method".---------------------------The Nelder-Mead (polytope) method is an ingeniously formulated series oftrials-and-errors.  Derivatives are not used and no assumption is madeabout the form of the objective function.  The method is therefore immuneto most of the pathological conditions affecting Gauss-Newton-Marquardt andGauss-Newton-SVD.  This property makes it useful for improving bad startingvalues prior to using a Gauss-Newton method, and for confirming the outcomeof a Gauss-Newton fit.  It can be used for the entire fit, but isdistressingly slow to converge if there are more than 3 or 4 parameters.The Gauss-Newton methods are usually much faster for least-squares fitting,because they "know" much more about the function to be fitted, whereasNelder-Mead knows only the values of the objective function.Like many others, I have tinkered with the algorithm but have not found away to overcome its main vice:  a tendency to premature convergence whilestill far from the solution.  This unfortunate behaviour occurs after aseries of early successes have expanded the polytope along some parameteraxes but not others.  A drastic change in shape of the polytope is thenrequired when the expansion reaches a valley floor.  In the course ofrearrangement the hitherto unexpanded parameters do not always expandquickly enough for the algorithm to notice that progress can be made downthe valley.  The only ways around this problem are to give good startingguesses and to specify a sufficiently small value of e [the convergencecriterion]----------------------------Robert D PurvesDepartment of PharmacologyUniversity of OtagoPO Box 913, DunedinNew Zealand---X-Sender: jelliffe.at.hsc.usc.eduDate: Mon, 24 Aug 1998 23:12:17 -0700To: PharmPK.-at-.pharm.cpb.uokhsc.eduFrom: Roger Jelliffe Subject: Re: Robustness of the Simplex methodMime-Version: 1.0Dear Craig:	I am not sure what you mean by robust. However, the Nelder-Mead simplexalgorithm seems never to blow up. It always gets at least a local minimum.There are no derivatives to calculate. It is slow, dumb, and plodding, butit gets there, and is safe, I believe, to use by someone not well versed infitting. This is why we have chosen it in our USC*PACK programs, forexample, for about 20 years. It is well discussed in Numerical Recipes,1986, Cambridge University Press, page 276, and pp. 289-293.Best regards,Roger Jelliffe************************************************Roger W. Jelliffe, M.D.USC Lab of Applied PharmacokineticsCSC 134-B, 2250 Alcazar St, Los Angeles CA 90033**Note our new area codes below, since 6/15/98!**Phone **(NOTE NEW AREA CODE AND PREFIX)** (323)442-1300, Fax (323)442-1302email=jelliffe.-at-.hsc.usc.edu************************************************You might also look at our Web page for announcements ofnew software and upcoming workshops and events. It ishttp://www.usc.edu/hsc/lab_apk/************************************************`
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