Solve Lagrange multipliers optimization problem

Question

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]

Answer 1

The short answer is that this is not a correct formulation of a Lagrange multiplier problem in the first place. In the question, derivatives are being taken with respect to the multipliers $\lambda1$ and $\lambda2$. That's simply wrong. These multipliers are there to enforce the fact that the gradients of the objective function and constraint conditions are linearly dependent. But they aren't themselves variables in the gradient.