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

• To: mathgroup at smc.vnet.net
• Subject: [mg55693] Re: [mg55670] Happy with v. 5.1.1. --- NMinimize, and MathOptimizer Professional
• From: "Janos D. Pinter" <jdpinter at hfx.eastlink.ca>
• Date: Sun, 3 Apr 2005 05:51:01 -0400 (EDT)
• References: <200504020628.BAA10750@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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:

NMinimize
{-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. http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/hundred.html
and
http://mathworld.wolfram.com/Hundred-DollarHundred-DigitChallengeProblems.html)
-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. (http://www.ijournals.net). 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.
Web: www.pinterconsulting.com

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```

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