I wrote some code, which might look messy, but hope you can see through it and tell me how to make it work. Minimize outputs itself, which is becouse it doesn't have enough information to solve the problem?
n = 2;
k = Binomial[n,2];
m = n + k;
Xs = Array[x,n];
Ds = Array[d,m];
PHIs = Array[phis,n-1];
Cs = Array[c,n];
c[1]=0
c[2]=0
x[1]=d[1]*Cos[d[3]]*Cos[phis[1]]-d[2]*Sin[d[3]]*Sin[phis[1]]+c[1]
x[2]=d[1]*Sin[d[3]]*Cos[phis[1]]+d[2]*Cos[d[3]]*Sin[phis[1]]+c[2]
expr = x[1]^2-x[2]-2
objfun = -Times @@ Part[Ds,1;;n]
Minimize[{objfun,{phis[1]>=0,phis[1]<=2*Pi,!Exists[phis[1],expr>0]}},{d[1],d[2],d[3]}]
The code supposed to solve:
min
$-d_1*d_2$
s.t
$x_1^2-x_2-2\leq 0$
where
$x_1=d_1*cos(d_3)*cos(\phi_1)-d_2*sin(d_3)*sin(\phi_1)+c_1$ and
$x_2=d_1*sin(d_3)*cos(\phi_1)+d_2*cos(d_3)*sin(\phi_1)+c_2$
for all
$\phi_1$ in the [0;2pi] interval, which can be transformed to the !Exist form according to Frank's suggestion
This problem is finding the maximal area ellipse in the
$x_2\geq x_1^2-2$ region, given it's center (0,0).
