Is group theory useful in any way to optimization?

Question

For what I have seen, optimization uses a lot of linear algebra and convex analysis, but I have not seen any group theory being used, so I was curious about it.

Please be so polite to give at least links to the other place when posting twice the same question. — j.p., May 15 '14 at 11:27

Olexandr Konovalov · Answer 1 · 2014-05-15T08:53:39.810

Yes, certainly. Please search for the papers with keywords "symmetry breaking constraint programming" to find out more information. In particular, see Symmetry in Constraint Programming by Ian P. Gent, Karen E. Petrie, Jean-François Puget, Chapter 10 in Handbook of Constraint Programming, Edited by F. Rossi, P. van Beek and T. Walsh, Elsevier, 2006.

Basically, knowing symmetries of solutions allows to reduce the search space: if it is known that the property of being a solution to the problem is preserved under some symmetries, then we have to check only representatives of orbits of solutions with respect to these symmetries, and groups naturally occur in studying this.

mesel · Answer 2 · 2014-05-15T08:30:28.223

5

Yes, One of the famaus one is that Rubik's Cube can be solved by at most $20$ move.

Since symetries of Rubik's cube forms a group, it is proved by the means of properties of Cayley graph of its group.

http://en.wikipedia.org/wiki/Optimal_solutions_for_Rubik%27s_Cube

It has also some applications in crypto systems. (To broke the codes faster by brute force)

edited May 15 '14 at 08:30

answered May 14 '14 at 21:39

mesel

15,125

I don't understand your Rubik's cube example. This is not an application of group theory to optimisation, but rather is an optimisation problem in group theory (of which there are many!). – user1729 May 15 '14 at 09:00
@user1729: please check the link, I did not get your objection. – mesel May 15 '14 at 11:12
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The question asks "Is group theory useful in any way to optimization?" So, to answer it we should find a problem from optimisation theory which uses group theory in the solution. This is not what your example does. Rather, your example is an optimisation problem from group theory. It is not an application of group theory, but rather is group theory. – user1729 May 15 '14 at 11:27
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@user1729: Rubik's cube is a toy,game and it is created as an puzzle. But we can use group theory to get useful result about solution of this puzzle. That is why I do not think Rubik's cube as a part of group theory.\
"Is group theory useful in any way to optimization?", I regard this question as Is there any optimization done by the means of group theory ? (I hope I do not get it wrong !)

I thougt it as an good applicaiton of group theory since many people thouht that group theory is too abstract and it has no applicaion in real life.
– mesel May 15 '14 at 11:43

score 2 · Answer 3 · answered May 14 '14 at 21:40

In the case of convex optimization, not really - you're playing with properties of convex functions and differentiability abilities (usually modifying the idea of gradient descent in deterministic or stochastic form).

However, for some types of combinatorial optimization, some group and field theory is useful (some arise in existence of certain types of codes, for example). A case where this is applied is in chemistry, where a lot of properties of molecules are encapsulated via symmetry groups and what not, so group theory provides some of the constraints for some function which needs to be optimized (such as energy of a configuration). There is also some interplay with algebraic geometry.

Is group theory useful in any way to optimization?

3 Answers3