Timeline for When does symmetry in an optimization problem imply that all variables are equal at optimality?

5 events

when toggle format	what		by	license	comment
Sep 7, 2011 at 20:55	history	edited	Dirk	CC BY-SA 3.0	Corrected wording
Mar 17, 2011 at 16:21	comment	added	Mike Spivey		Thanks, Dirk; now I see what you mean. That's an interesting generalization.
Mar 17, 2011 at 9:39	comment	added	Dirk		Maybe I just illustrate with your example: The symmetry you had in your question is the symmtery under the action of the permutation group. The optimization variable is $(x,y)$ and the objective $f(x,y) =x+y$ does not change under the mapping $P(x,y)= (y,x)$, i.e. $f(x,y) = f(y,x)$. Moreover, the mapping $P$ maps the feasibile region $x^2+y^2\geq 1$, $x,y\geq 0$ one-to-one and onto itself. Consequently, if $(x^,y^)$ is a solution, $P(x^,y^) = (y^,x^)$ also is.
Mar 17, 2011 at 9:19	comment	added	Mike Spivey		Dirk, your answer sounds interesting. Would you mind elaborating, for my sake? For instance, I don't think I could look at an optimization problem and recognize when your criterion applies. (My background is operations research, and my abstract algebra is unfortunately quite rusty.)
Mar 17, 2011 at 8:03	history	answered	Dirk	CC BY-SA 2.5