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Re: FindMinimum

• To: mathgroup at smc.vnet.net
• Subject: [mg22183] Re: [mg22112] FindMinimum
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 17 Feb 2000 01:24:18 -0500 (EST)
• References: <200002160734.CAA17847@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

Johannes Ludsteck wrote:
> 
> Dear Mathgroup Members,
> I have to minimize a function which is continuous but not smooth.
> Someone suggested to use the flexible polyhedron search method
> by Nelder and Mead.
> The FindMinimum[] function in Mathematica doesn't use this
> method, but seems to do the job well if I force use of the secant
> method by givind two starting points for each variable.
> Since the features of FindMinimum are not documented very well, I
> wonder whether FindMinimum guarantees to find at least a local
> minimum in my case.
> 
> In case you are interested in the function:
> 
> Plus @@ Abs[y - max[Dot[x, b], 0]],
> 
> where y is a vector, x a matrix and b a vector. b contains the
> miminization arguments.
> 
> Johannes Ludsteck
> Centre for European Economic Research (ZEW)
> Department of Labour Economics,
> Human Resources and Social Policy
> Phone (+49)(0)621/1235-157
> Fax (+49)(0)621/1235-225
> 
> P.O.Box 103443
> D-68034 Mannheim
> GERMANY
> 
> Email: ludsteck at zew.de


The secant method will generally work fine in this situation.
Alternatively you might provide an explicit Gradient->... to FindMinimum
because it cannot compute this symbolically. Yet another alternative
would be to minimize an L2 rather than L1 distance (if this is a
reasonable thing to do for your problem) by summing squares rather than
absolute values.

Nelder-Mead is probably not the best approach given that it is a bit
crude. That said, we have an optimization package in development that
incorporates N-M as a method. I am hopeful that it will become a
standard add-on package once we're done polishing it. Possibly we will
also make it a method for FindMinimum, I'm not sure at this time.

Daniel Lichtblau
Wolfram Research

• References:
  • FindMinimum
    • From: "Johannes Ludsteck" <ludsteck@zew.de>