# Generalisation of dualities, what concept do dualities represent?

Duality is a concept that pops up in different areas of mathematics as well as other science, but besides being a "woo isn't that nice?", is there anything more to duality (than loosely stated some type of isomorphic behaviour)?

Has anyone seen duality (trippliality?? or more) between more than two conceptual objects?

Edit : By "anything more" I mean a systematic study and categorisation of duality on its own, maybe a chapter in a book. For example the properties of holomorphic functions are studied separate of the specific holomorphic functions themselves, so one can look at holomorphic functions as a subject of study on it's own right. Looking for something similar regarding the concept of duality, not the specific objects having duality (or triality), can we have a definition of duality independent of specific examples? In other words can we say this is what is meant by a duality relationship (or triality) and show that specific examples actually satisfy the definition and hence they have duality relationship wrt each other?

Apologies for a very ill defined question, but I wouldn't even know what area does a question like this belongs to.

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en.wikipedia.org/wiki/Triality –  t.b. Sep 12 '11 at 11:47
I don't understand what you mean by "anything more." What examples do you have in mind? –  Qiaochu Yuan Sep 12 '11 at 16:12
@Qiaochu : By anything more I mean a systematic study and catehorisation of duality on it's own, maybe a chapter in a book. For example the properties of holomorphic funstions are studied separate of the specific holomorphic funstions themselves, so one can look at holomorphic funstions as a subject of study itself. Looking for something similar regarding the concept of duality, but not the spefcific objects having duality (or triality), can we have a defintion of duality independent of specific examples? –  Arjang Sep 12 '11 at 21:11
@Arjang: I think "duality" has a sufficiently different meaning in enough different contexts that you're not going to be able to find an overarching theory of duality, although I would love to be proved wrong! –  Mike Spivey Sep 12 '11 at 21:18
For the modified question, perhaps mathoverflow.net/questions/73711/the-concept-of-duality will be useful. –  Qiaochu Yuan Sep 12 '11 at 21:52

It looks like you have two questions here. This answer addresses the one about uses of duality. (I'm interpreting "is there anything more to duality?" as "Are there any practical applications of duality?")

Not only is duality the most important theoretical concept in linear programming, it has a large number of useful applications. For example:

1. Any feasible solution to the dual problem gives a bound on the optimal solution to the original problem.
2. Related to that, if the dual problem is unbounded then there is no solution to the original problem (i.e., the original problem is infeasible).
3. Basic sensitivity analysis on the original problem (i.e., what happens to the optimal solution if we make small changes to the parameters in the original problem) can be done immediately by looking at the values of the optimal dual solution.
4. If the dual problem is easier to solve than the original problem, you can determine the optimal solution to the original problem by solving the dual instead.
5. For the transportation and assignment problems, two of the most important classes of linear programming problems, there are algorithms that exploit duality so that these problems can be solved much faster than when using the standard simplex method.
6. There are some algorithms for solving general linear programs that work with both the original and the dual problem simultaneously. See, for example, this answer on the primal-dual method. Wikipedia also claims that "the class of primal-dual path-following interior point methods is considered the most successful" of the interior-point methods (which are the only major competitors to the simplex method for solving general linear programs).
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@Arjang: I'm wondering, now that I've seen your comment, if I gave you the kind of answer you were looking for. If this isn't helpful, let me know, and I'll delete my answer. –  Mike Spivey Sep 12 '11 at 21:19
1 upvote was from me, hoping there would be more answers before I artificially accept only one as "the answer". I never belived answers having maximal chains, only that they complement each other. No please do not delete your answer as it is not just only intresting for me but for anyone else who ponders about duality and lands here. –  Arjang Sep 13 '11 at 2:24