I have two nested solid figure, where $V(a,h,\tau)$ defines the volume and $A(a,h,t)$ defines the surface. The outer solid figure is parametrized in $a_s$,$h_s$ and $t_s$ (they share a common center). Now I raise the Volume of the inner by factor $\alpha$, so
$$V(a_s,h_s,t_s) = \alpha V(a,h,t)$$
The first constraint is that the surface $A$ of the inner raised solid figure is constant: $$A(a,h,t) = A(a_s,h_s,t_s)$$
The second constraint is that the enclosed volume of outer and inner solid figure is constant:
$$V(a_sl, h_s+d, ts) - V(a_s,h_s,t_s) = V(al, h+d, t) - V(a,h,t)$$
If $\alpha$ gets too big, the surface constraint can not be satisfied any more. So I want to maximize this with respect to $\alpha$. The target function has to be $\alpha V(a,h,t)$.
I'm taking the vector derivative in all my parameters (except for $al$, which is the function alipid) and try to solve it symbolically. This runs forever... (several hours by now). Can I check the "progress" somehow? Or is this simply an ill-posed system?
Perhaps I should try to optimize this numerically with some starting values for $a$, $h$ and $t$? Finding good starting values will be difficult, as the system has lots of solutions.
d = 0.1;
V[a_, h_, tau_] := (Sqrt[3]/2) h (Sqrt[3] a + h/3 tau)^2
A[a_, h_, tau_] := 3 Sqrt[3] a^2 + 2 a h Sqrt[1 + tau^2] +
h (2 a + Sqrt[3]/3 h tau) Sqrt[4 + tau^2]
alipid[a_, h_, tau_, d_] := a + d Sqrt[3]/
6 ((1 + Sqrt[1 + tau^2] - tau )/(1 + Sqrt[1 + tau^2] + tau ) +
(2 + Sqrt[4 + tau^2] - tau)/(2 + Sqrt[4 + tau^2] + tau))
eqns := {α V[a, h, t] + λ1 (A[as, hs, ts] - A[a, h, t]) +
λ2 (V[alipid[as, hs + d, ts, d], hs + d, ts] - V[as, hs, ts] -
V[alipid[a, h + d, t, d], h + d, t] + V[a, h, t])}
deriv := D[eqns, {{α, as, hs, ts, λ1, λ2}}]
Solve[deriv == 0, {α, as, hs, ts, λ1, λ2}, Reals]
Manipulate
for variation parameters? $\endgroup$N@Simplify[deriv /. {t -> 0, a -> -1, h -> 1}, Assumptions -> {ts > 0, hs > 0, as > 0}]
and thenNSolve
that. I just made those values for $h$ etc up. $\endgroup$