Get minimum values for parameters of the following equations using W|A?

Questions
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1

Hello community,

I try to get the optimized values for a,b,c,d and e,f,g,h at the minimum of t of the following equations:

minimum t 
t = 0.5*(a+1)*a/e + 0.75*(b+1)*b/f + (c+1)*c/g + 1.25*(d+1)*d)/h,
(2+a*z)*(2+b*z)*(2.5+1.5*c*z)*(3+2*d*z)*100 = 10000, 
e+f+g+h = 4000000,
z = SQRT(2600)/50000,

Wolfram doesn’t understand what I want, so either it’s to complex or I have some problems to find the right syntax.
Each part of the first function is strictly monotone and the summation has an upper bound (u is fix).
The second equation contains the values a to d which are end values of simple arithmetic sums [e.g. 1/2(x+1)x] from the first equation. I’m pretty sure, there should be only one global minimum.

Any ideas to get a solution?

2

Hi Mike,
You need to follow the Mathematica documentation regarding capital letters, ==, instead of =, avoid c and e as variables, etc… Below I modified slightly your code and solved:

NMinimize[{t, 
  t == 0.5*(a + 1)*a/\[Epsilon] + 
    0.75*(b + 1)*b/f + (\[Kappa] + 1)*\[Kappa]/g + 
    1.25*(d + 1)*
     d/h, (2 + a*z)*(2 + b*z)*(2.5 + 1.5*\[Kappa]*z)*(3 + 2*d*z)*100 ==
    10000, \[Epsilon] + f + g + h == 4000000, 
  z == Sqrt[(2600)/50000]}, {a, \[Epsilon], z, f, g, h, b, \[Kappa], 
  d, t}]

{-6.7612, {a -> 2.90222, \[Epsilon] -> 7.25383, z -> 0.228035, 
  f -> 3.99999*10^6, g -> 2.58406, h -> -2.47564, 
  b -> 2.37504, \[Kappa] -> 1.77412, d -> 3.85404, t -> -6.7612}}