Maximize an expression with respect to a variable after its minimization with respect to other variables

Question

I started to use Mathematica a few time ago. I want to minimize the following expression (function of $l,p,q,r,c$) with respect to variables $l, p, q, r$ and then maximize the result obtained with respect to variable $c$. However, when I try to obtain an expression function of $c$ to maximize later using Minimize, I do not get any result because it takes too long. How can I solve this issue?

Minimize[{(((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4+c))+(l*r)*(1-1/2)+(p^2/2)*(1-c)+(p*q)*(1-2*c)+(p*r)*(1-(1/4+c))+(q^2/2)*(1-c)+(q*r)*(1-1/4)+(r^2/2)*(1-(1/4-c)))/((l^2/2)*(1-(1/4-c))+(l*p)*(1-1/4)+(l*q)*(1-(1/4-c))+(l*r)*(1-0)+(p^2/2)*(1-c)+(p*q)*(1-(1/4-c))+(p*r)*(1-0)+(q^2/2)*(1-(1/4-c))+(q*r)*(1-1/4)+(r^2/2)*(1-0))), l >= 1, p >= 0, q >= 0, r >= 0, l + p + q + r == 1000000, 1/7<c<1/5}, {l, p, q, r}]

Why does the above command never end?

Answer 1

This can be done numerically (only numerically in my opinion) in a standard way which takes a lot of time:

f[c_?NumericQ] := NMinimize[{(((l^2/2)*(1 - (1/4 - c)) + (l*p)*(1 - 1/4) + (l*
       q)*(1 - (1/4 + c)) + (l*r)*(1 - 1/2) + (p^2/2)*(1 - 
       c) + (p*q)*(1 - 2*c) + (p*r)*(1 - (1/4 + c)) + (q^2/2)*(1 -
        c) + (q*r)*(1 - 1/4) + (r^2/2)*(1 - (1/4 - c)))/((l^2/
       2)*(1 - (1/4 - c)) + (l*p)*(1 - 1/4) + (l*
       q)*(1 - (1/4 - c)) + (l*r)*(1 - 0) + (p^2/2)*(1 - c) + (p*
       q)*(1 - (1/4 - c)) + (p*r)*(1 - 0) + (q^2/
       2)*(1 - (1/4 - c)) + (q*r)*(1 - 1/4) + (r^2/2)*(1 - 0))), 
l >= 1, p >= 0, q >= 0, r >= 0, l + p + q + r == 1000000 
}, {l, p, q, r}, Method -> "DifferentialEvolution"][[1]]
f[0.196]
(*0.732468*)
Plot[f[c], {c, 0.19, 0.20}]

NMaximize[{f[c], 1/7 <= c && c <= 1/5}, c, Method -> "DifferentialEvolution"]
(*{0.733816, {c -> 0.2}}*)