DSE 2012 Q45

[Dreyfus]

According to me the function f(x,y) is convex and if this function is convex then its epigraph ie L={(x,y)ER² | f(x,y)<=c} for constant c must be Convex set.
But answer key says set U is convex set.
Please help me where I m missing.....

[Arushi :))]

in the diagram it should be c.. wrote c+1 by mistake

[Dreyfus]

Arushi.....the curves that you drawn are the level curves of the function f(x,y). It is not possible to draw function as it is defined in R³ space......so we cannot say anything about convex set merely on the basis of level curves!

[Dreyfus]

And I used convex combination method to determine that the function in convex... See pic below

[SoniaKapoor]

check this,duck explaination
http://discussion-forum.2150183.n2.nabble.com/dse-2012-q45-please-help-td7582382.html#a7582443

[Dreyfus]

Sonia...I saw neha 's explanation but according to the snapshot ( from AC Chiang) if the function is convex then the convex set must be L !
Also it is given that this case is equally valid for n variable case! That's y I m confused!

[SoniaKapoor]

I'm not getting conves.I used hessian but no definite conclusion can be reached from it.Best is to draw for such quest.

[Dreyfus]

Yes hessian won't be helpfull in this case...that's y I used convex combination method and I got this function as convex....

[SoniaKapoor]

A convex function and convex set are 2 diff things. A function which is convex need not be convex set.

[Dreyfus]

Yes I know this Sonia......I m not saying that every convex function possess convex set.......its better u see the highlighted area in the above snapshots and I hope  u'll get what I m trying to say.....

[Granpa Simpson]

Vaibhav ur doubt is a valid one..seems quite confusing yarrrr...anybody plz help on this..even i have the same doubt...

[Nupur]

Yeah! I did not have a doubt in this question until now! But now I do! But no such thing (as you've shown in Chiang) is written in Hammond! Amit sir please chip in!!

[Amit Goyal]

f(x, y) = x + y + xy
can be written as
f(x, y) = (x + 1)(y + 1) - 1
If you plot the level curves in (x, y) space, you can easily see that level curves are convex (and not f(x, y)) which along with the fact that the function f(x, y) is increasing in both x and y, gives us quasi-concavity of f(x, y). Thus, U is a convex set.

Alternatively, you may rule out the wrong options.
Clearly, L = {(x,y)| f(x, y) <= c} is not a convex set because
if we consider two points (0, 1) and (1, 0) in L for c = 1, then for t = 0.5
we get f(t (1, 0) + (1-t) (0, 1)) = f(0.5, 0.5) = 1.25 > c = 1.

And I = {(x,y)| f(x, y) = c} is also not a convex set because
if we consider the same example as above i.e. two points (0, 1) and (1, 0) in I for c = 1, then for t = 0.5
we get f(t (1, 0) + (1-t) (0, 1)) = f(0.5, 0.5) = 1.25 > c = 1.

[Dreyfus]

Amit Sir thanks a lot for the reply. As you mentioned function f(x,y) is not convex but if I use Jensen's inequality ie the convex ccombination method then I am getting this function as convex ( my real confusion as if its convex then L must be convex set). I have attached my workings please guide me Sir where I am going wrong.IMG_20140616_122758_1.jpg