Unfortunately, even though the (entropy) function is
very-well behaved, symbolic solution is impossible for any
but the simplest problems. In this particular case, I was
just trying the augmented Lagrangian method, and I thought I
would start with a rather toy problem.
A more typical problem for what I'm trying to do would be an
h(x_1,..,x_20), with 5 or 10 linear constraints.
Kostas
Stavros Macrakis wrote:
> On 5/18/07, *Kostas Oikonomou* <ko at research.att.com
> <mailto: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
>
>