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In article <C1F8H9.8nw@world.std.com>, jbk@world.std.com (Jeffrey B Kane) writes:
|> 
|> I was wondering if anyone could point me to some example code and/or references
|> for different penalty functions to be used in optimization problems.  Although
|> a number of sources make somewhat vague references to these functions 
|> (Goldberg, Davis, and others in their books) I wanted some concrete examples 
|> that I could look at.
|> 
|>        Thanks,
|>              Jeffrey
|> 
|> 
|> ======================================
|> Jeffrey Kane, MD
|> Kane Biomedical Systems
|> Boston, MA
|> 
|> Internet    jbk@world.std.com
|> Compuserve  74206,640
|> AppleLink   D0738
|> 
|> [Don't take life too seriously... it's not like anyone gets out of it alive.]

Multiplier methods, for constrained optimization problems, generally
exceed the performance of penalty function methods.  The reference:
R.A. Tapia, "Diagonalized Multiplier Methods and Quasi-Newton Methods for
Constrainded Optimization", Journal of Optimization Theory and Applications,
Vol. 22., No.2, June 1977, p. 135-194., is where I got started with these
methods.  Two papers on Penalty Function methods are in the reference section
of this paper.  They should point you to some examples of penalty functions.

Doug Hart
hart@de01.denver.waii.com

hart@de01.denver.waii.com said:
>
>Multiplier methods, for constrained optimization problems, generally
>exceed the performance of penalty function methods.  The reference:
>R.A. Tapia, "Diagonalized Multiplier Methods and Quasi-Newton Methods for
>Constrainded Optimization", Journal of Optimization Theory and Applications,
>Vol. 22., No.2, June 1977, p. 135-194., is where I got started with these
>methods.  Two papers on Penalty Function methods are in the reference section
>of this paper.  They should point you to some examples of penalty functions.


Multiplier methods are fine for *continuous* optimization, but not readily
applicable for many combinatorial problems, unless a convenient continuous
relaxation exists.  Multiplier methods also require that you have an explicit
representation of the constraint set in a particular form.  For example, in
a Travelling Salesman problem the logically simply constraint "Each tour must
pass through each city exactly once" translates into an exponential number of
linear inequalities in the linear/integer programming formulation.

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