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Re: Re: Happy with v. 5.1.1. --- NMinimize, and MathOptimizer Professional


For clarity, let me just add a positive note: after increasing the global 
search effort (from its the default value), MathOptimizer and MathOptimizer 
Pro both find the solution in Trefethen's 4th problem to the required 
10-digit accuracy.

This fact is noted also in the book "The SIAM 100-Digit Challenge: A Study 
in High-Accuracy Numerical Computing", by Folkmar Bornemann · Dirk Laurie · 
Stan Wagon · Jörg Waldvogel. See the site includes 
Mathematica code for this problem, as well as for the others.


Janos D. Pinter
PCS Inc.
E-mail: jdpinter at

At 06:51 AM 4/3/2005, Janos D. Pinter wrote:

>At 02:28 AM 4/2/2005, Skirmantas wrote:
> >I'm greatly pleased with the new version of NMinimize in Mathematica
> >5.1.1. For some reason, NMinimize was working worse in 5.1.0 than in
> >5.0.0. (and I complained about it here). Now, in 5.1.1., it again produces
> >excellent results. It is more than a minor upgrade; your 5.1.1. NMinimize
> >may produce different (hopefully, better) results than 5.1.0. NMinimize.
>Skirmantas and others who may be interested,
>Indeed, NMinimize has been greatly improved since its introduction in v.
>4.2 (if I recall well). At the same time, let me point out that the model
>class addressed by NMinimize (mixed integer nonlinear optimization)
>includes very difficult problems, and no solver is perfect to handle 'all'
>such models, especially not so in its default operational mode.
>To illustrate this point, consider e.g. Trefethen's Problem 4
>minimize Exp[Sin[50*x]] + Sin[60*Exp[y]] + Sin[70*Sin[x]] + Sin[Sin[80*y]]
>- Sin[10*(x + y)] + (x^2 + y^2)/4
>(w/o stating explicit variable bounds).
>Solving this model by NMinimize and the third party application
>MathOptimizer Professional (both used in default solver mode), and applying
>the tentative variable bounds [-1,1], [-10,10], [-100,100], one can verify
>that 1) neither solves the model to the known global optimum, and 2)
>MathOptimizer Professional consistently finds a better quality solution
>than NMinimize, for this particular problem.
>The solutions found are:
>{-2.8503166859007303, {x -> -0.02541875771380319, y -> 0.2901752668435903}}
>{-2.2781355618824097, {x -> -1.5338718828334104, y -> 0.36797859719983905}}
>{-1.8567821476829507, {x -> 0.34251420810742605, y -> 0.5084031754711565}}
>MathOptimizer Professional
>{-3.1440794103113316, {x -> -0.0231677651, y -> -0.4942128768}}
>{-3.1440794103103786, {x -> -0.0231677607, y -> -0.4942128585}}
>{-2.966665564173799, {x -> 0.242806842, y -> -0.0933238715}}
>The true global solution is
>(see e.g.
>-3.306868647 4752372800 7611377089 8515657166...
>Interested colleagues may like to check out our paper, written with Frank
>Kampas, that just appeared in Mathematica in Education in Research 10
>(2005) 2, 1-18. ( In this work, we present a
>systematic comparisons between NMinimize and MathOptimizer Professional in
>solving several (uniform and non-uniform size) circle packing models. These
>and similar difficult global optimization problems can pretty well and soon
>'humble' any numerical optimization package, as the model-size (here: the
>number of circles) increases.
>Janos D. Pinter
>PCS Inc.
>E-mail: jdpinter at
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