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}}