       Re: FindMinimum

• To: mathgroup at smc.vnet.net
• Subject: [mg8259] Re: FindMinimum
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sat, 23 Aug 1997 01:58:56 -0400
• Organization: University of Western Australia
• Sender: owner-wri-mathgroup at wolfram.com

```Fahad A Hoymany wrote:

> Here is the problem in case some kind soul wants to give it a try:
>
>   Find the minimum of f in c and h
>           f = (c d1+d2) r (2/((c+1) (h-c+1))) + d3 h/2 +
>               (c d4+d5) r (2/((c+2)*(h-c+1)))
>
>   Note that c <= h, and c,h >=0.  If it helps, h < 10,000.  The d's are
> real numbers in the range 0 to 1.0, and r is real in the range 0 to
> 100,000.

Here is your function rewritten to use subscript notation:

In:= f[c_, h_] := (2*(c*Subscript[d, 1] + Subscript[d, 2])*r)/
((c + 1)*(-c + h + 1)) + (h*Subscript[d, 3])/2 +
(2*(c*Subscript[d, 4] + Subscript[d, 5])*r)/((c + 2)*(-c + h + 1))

To find the extrema you can compute the partial derivatives with respect
to x and h. You only need to set the Numerator:

In:= {Numerator[Together[D[f[c, h], c]]] == 0,
Numerator[Together[D[f[c, h], h]]] == 0} /.
{r -> Random[Real, {0, 100000}],
Subscript[d, _] :> Random[Real, {0, 1}]}

Out=
4            3              2            2
{2 (13011.9 c  + 102386. c  - 18052.9 h c  + 272405. c  +

50150.4 h c + 309163. c + 2736.01 h + 72338.6) == 0,

4              3             3
0.581759 c  - 1.80445 h c  + 0.653476 c  +

2  2              2            2
0.986612 h  c  - 3.38305 h c  - 31200.5 c  +

2                                        2
2.591 h  c + 1.9914 h c - 207997. c + 0.107197 h  +

0.303971 h - 132371. == 0}

I have substituted in numerical values for r and the ds. You can also
solve the equations algebraically yielding some very large polynomial
expressions.

You can now use NSolve

In:= NSolve[%, {h, c}]

Out= {{h -> -257.165, c -> 2.44962},
{h -> -180.422 - 36.6747 I, c -> -3.51691 - 16.5083 I},
{h -> -180.422 + 36.6747 I, c -> -3.51691 + 16.5083 I},
{h -> -44.721 - 2056.97 I,  c -> -0.0535149 - 0.000527709 I},
{h -> -44.721 + 2056.97 I, c -> -0.0535149 + 0.000527709 I},
{h -> -1.02774, c -> -0.712571},
{h -> 2.83194, c -> -5.96283},
{h -> 137.698, c -> -18.607},
{h -> 214.547, c -> 10.6122}, {h -> 243.964, c -> 3.53549}}

I tried several random values and found none that satisifed your
constraints.  Also, you have to check whether these are maxima, minima,
or inflection points.

Cheers,
Paul

____________________________________________________________________
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA  6907                     mailto:paul at physics.uwa.edu.au
AUSTRALIA                              http://www.pd.uwa.edu.au/Paul

God IS a weakly left-handed dice player
____________________________________________________________________

```

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