In the context of optimization algorithms, what is archimedean weighting? Especially examples are welcome :)

(Thanks to Paul for bringing this keyword to my attention)

asked 18 Feb '13, 08:56

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Geoffrey De ... ♦
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...as found in this question?

(18 Feb '13, 09:59) fbahr ♦

@fbahr yes, for example as found there, but doesn't have to relate to multi-objective optimization necessarily. I am looking for a clear, unambiguous example and/or definition.

(18 Feb '13, 10:05) Geoffrey De ... ♦
1

See here for another note from Paul. To me it seems more like a naming convention.

(18 Feb '13, 12:33) Ehsan ♦

In short: "Slapping weights on each objective [of a multi-objective program, edit] and adding them up."

Here, afaiu, @Paul refers to weighted goal programming, in which weights are assigned to deviations of objectives from their perspective goal.

Let's say, you have a program:

\[ \begin{align} f_1(x) &\ge v_1\\ f_2(x) &= v_2\\ f_3(x) &\le v_3\\ x &\in X \end{align} \]

Then, its "Archimedian" formulation is:

\[ \min \alpha_{s_1^-} \times s_1^- + \alpha_{s_2^-} \times s_2^- + \alpha_{s_2^+} \times s_2^+ + \alpha_{s_3^-} \times s_3^+\\ \begin{align} f_1(x) &+ s_1^- & &\ge v_1\\ f_2(x) &+ s_2^- &- s_2^+ &= v_2\\ f_3(x) & &- s_3^+ &\le v_3\\ s &\ge 0, & x &\in X\\ \alpha_i & > 0, & \sum \alpha_i &= 1 \end{align} \]

link

answered 18 Feb '13, 10:06

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fbahr ♦
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edited 19 Feb '13, 07:06

Does archimedean weighting imply the use of deviations of objectives from their perspective goal? Stated differently, does it imply that if v1 = 10 that a solution with f1(x) = 5 should have a worse weight than another solution with f1(x) = 8?

(18 Feb '13, 10:15) Geoffrey De ... ♦

Err, I failed to put the most important part of the "Archimedian" formulation in my answer above: in weighted goal programming, the \(\alpha_i\)'s (generally) can take any value (\(> 0\)); a weighting is called "Archimedian" iff (also) \(\sum \alpha_i = 1\).

(18 Feb '13, 10:31) fbahr ♦

So, whether a solution with \(f_1(x) = 5\) has a worse weight than an(y )other solution with \(f_1(x) = 8\) may depend on your choice of \(\alpha_1\) – but since \(\alpha_1\) is a positive constant, the "penalty" for having \(f_1(x) = 5\) in your solution is always higher than for \(f_1(x) = 8\) [may be over-compensated by \(f_2, f_3\), though].

(18 Feb '13, 10:48) fbahr ♦
2

For what little it's worth, I usually see "Archimedean" used in the context of goal programming (which implies weighting deviation variables), but as far as I know the term can also be applied to taking a weighted linear combination of objective values to create a single criterion function.

Non-archimedean weighting, in the context of goal programming, appears to be synonymous with "preemptive priorities".

(18 Feb '13, 18:12) Paul Rubin ♦
1

@Paul: As I told you before (and tell you again) - things would be so much easier for us if we had something like "Paul's Compendium of Q&As in the World of OR".

Whenever a "new" question pops up, chances are that you already answered them somewhere, somewhere, seomewhy (on sci.op-research, IBM's opt. forums, etc.pp)!

> sudo write-it!

;-)

(19 Feb '13, 04:02) fbahr ♦

@fbahr: A couple of things to consider: (a) I can't recall when or where I've answered things, so it would be hard to assemble a compendium; (b) not all my answers are correct (and hence not all bear repeating). In fact, I can't swear that a majority of my answers have been correct. I learned along time ago that saying something with an air of certainty often trumps saying the correct thing. :-)

(20 Feb '13, 11:15) Paul Rubin ♦
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Asked: 18 Feb '13, 08:56

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