On 5/18/07, Kostas Oikonomou <ko at research.att.com> wrote:
>
> I am trying to use augmented_lagrangian_method() to minimize
> the function
>
> h(x,y,z,w) := x*log(x) + y*log(y) + z*log(z) + w*log(w)
>
> where x,y,z,w > 0, subject to the constraints x+y+z+w=1 and
> 3x + 10y = 2.
>
> This is a strictly convex function over a convex domain, so
> there is a unique global minimum, and in this case it is
> approximately
>
I was wondering if I could solve this symbolically using a sloppy but simple
approach...
I used the constraints to reduce the number of variables from 4 to 2.
Solve for zeroes of dh/dx and dh/dw. Equate them. You get
(5*w-2)^(3/7)*(8*2^(6/7)-20*2^(6/7)*w)+7*w*(6*w-1)^(3/7) = 0
This is equivalent to the polynomial
10239822114712*w^10-40959911057356*w^9+73727985176226*w^8-78643199176457*w^7\
+55050240000000*w^6-26424115200000*w^5+8808038400000*w^4-2013265920000*w^3+3019898880\
00*w^2-26843545600*w+1073741824
Unfortunately, this doesn't factor, but all the steps up to now have been
exact, and its zeroes can easily be found to any desired precision using
realroots. To 30 digits, we get
w = 0.315752406058113692714082957681
w = 0.557258532275860786576382557583
The second solution is spurious. Back substitute to get the other
variables.
-s