Re: [eclipse-clp-users] Pareto Optimality + Branch and Bound?

"However, it is impossible to obtain points on non-convex portions of the
Pareto front"

OP said that single element of Pareto Point would be enough. For non-convex
Pareto sets and "smooth" traversing Pareto sets see, for example

http://www.iiasa.ac.at/Admin/PUB/Documents/WP-79-066.pdf

A.L.


-----Original Message-----
From: Marco Zimmerling [mailto:marco_at_...260...] 
Sent: Thursday, April 29, 2010 3:38 AM
To: eclipse-clp-users_at_lists.sourceforge.net
Subject: Re: [eclipse-clp-users] Pareto Optimality + Branch and Bound?

Hi Philipp,

I'm not aware of any ECLiPSe framework for Pareto front 
generation/approximation.

As Andrzej suggested, one of the most common methods is the weighted sum 
approach. Assuming two objective functions f1(X) and f2(X), with X being 
the vector of decision variables, the single criteria problem becomes

   f(X) = w1*f1(X) + w2*f2(X),

where w1 and w2 are weights that *should* reflect the importance of each 
objective. Typically, one assumes w1 + w2 = 1. By solving this problem 
for different combinations of w1 and w2, you can generate different 
Pareto optimal points. However, it is impossible to obtain points on 
non-convex portions of the Pareto front, and it is often difficult to 
generate a smooth Pareto front (i.e., without holes). So it is 
worthwhile to look into other approach as well. The article "Survey of 
multi-objective optimization methods for engineering" by Marler and 
Arora (Struct Multidics Optim 26, pp. 369-395, 2004) provides a good 
overview of multiobjective optimization methods, including a discussion 
of pros and cons.

Another branch of research uses evolutionary methods to find good 
approximations of the Pareto front. The optimization problem is treated 
as a black box; the focus lies on efficient stochastic search methods. 
If you want to get a feeling: 
http://www.tik.ee.ethz.ch/sop/publicationListFiles/zlb2004a.pdf

Marco

Andrzej Lewandowski wrote:
> Convert to standard optimization problem (i.e. single criteria) using
> weighted sum of individual criteria
> 
> A.L.
> 
> -----Original Message-----
> From: Philipp Marcus [mailto:marcus_at_...247...] 
> Sent: Wednesday, April 28, 2010 4:13 PM
> To: eclipse-clp-users_at_lists.sourceforge.net
> Subject: [eclipse-clp-users] Pareto Optimality + Branch and Bound?
> 
> Hi,
> 
> I have a COP with two cost variables and I'm searching for a pareto 
> optimal solution. Though it would be nice to get the complete pareto 
> front, I'd already be happy with *one* pareto optimal value. Is there a 
> framework I could use? If not so, do you have any suggestions how this 
> could be implemented?
> 
> Best regards,
> Philipp
> 
>
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