I’m new to Mathematica and am searching for a approach to make use of Decrease/Maximize to unravel for a tough optimization downside.

I’ve solved for the worldwide optimum for a given $n$ numerically and am hoping for a proof of $$ maxleft( {lim_{ntoinfty}f}proper)=1.$$

The perform I’m thinking about maximizing is within the following kind:

$$displaystyle f(x_1,x_2, …,x_n,y_1,y_2, …,y_{n-1}) =a_1 b_1 prod_{i=2}^na_i^3$$

the place

$$displaystyle a_i = start{instances} frac{1+x_i}{1+y_i+sqrt{(1-y_i)^2+y_i(1-1/x_i)^2}} &textual content{for }i=1,2,…,n-1 (1+x_n)/2 &textual content{for }i=n finish{instances}$$

and

$$displaystyle b_i=start{instances} (1-x_i)left[frac{1+x_i-2y_i a_i}{left(1-frac{y_i}{x_i}a_iright)^2}right] &textual content{for }i=1,2,…,n-1 (1-x_n)(1+x_n) &textual content{for } i=n finish{instances}

$$

topic to $n-1$ constraints:

$$displaystyle b_2=b_1 y_1 a_2^2, b_3= b_1 y_1 a_2^2 y_2 a_3^2, … , b_j=b_1prod_{i=2}^{j} a_i^2 y_{i-1} textual content{for } j=2,3… , n $$

and

$$displaystyle zero leq x_i leq 1, zero leq y_i leq 1.$$

For instance, the next Mathematica code solves for n=3:

```
a1 = (1 + x1)/(1 + y1 + Sqrt[(1 - y1)^2 + y1*(1 - 1/x1)^2]);
a2 = (1 + x2)/(1 + y2 + Sqrt[(1 - y2)^2 + y2*(1 - 1/x2)^2]);
a3 = (1 + x3)/2;
b1 = (1 - x1)*(1 + x1 - 2*y1*a1)/(1 - y1/x1*a1)^2;
b2 = (1 - x2)*(1 + x2 - 2*y2*a2)/(1 - y2/x2*a2)^2;
b3 = (1 - x3)*(1 + x3);
fmax = b1*a1*a2^3*a3^3;
constraint1 = b2 == b1*y1*a2^2 ;
constraint2 = b3 == b1*y1*a2^2*y2*a3^2;
NMaximize[{fmax, constraint1, constraint2,
0 <= {x1, x2, x3, y1, y2} <= 1}, {x1, x2, x3, y1, y2}]
```

and the next code solves for n=4:

```
a1 = (1 + x1)/(1 + y1 + Sqrt[(1 - y1)^2 + y1*(1 - 1/x1)^2]);
a2 = (1 + x2)/(1 + y2 + Sqrt[(1 - y2)^2 + y2*(1 - 1/x2)^2]);
a3 = (1 + x3)/(1 + y3 + Sqrt[(1 - y3)^2 + y3*(1 - 1/x3)^2]);
a4 = (1 + x4)/2;
b1 = (1 - x1)*(1 + x1 - 2*y1*a1)/(1 - y1/x1*a1)^2;
b2 = (1 - x2)*(1 + x2 - 2*y2*a2)/(1 - y2/x2*a2)^2;
b3 = (1 - x3)*(1 + x3 - 2*y3*a3)/(1 - y3/x3*a3)^2;
b4 = (1 - x4)*(1 + x4);
fmax = b1*a1*a2^3*a3^3*a4^3;
constraint1 = b2 == b1*y1*a2^2 ;
constraint2 = b3 == b1*y1*a2^2*y2*a3^2;
constraint3 = b4 == b1*y1*a2^2*y2*a3^2*y3*a4^2;
NMaximize[{fmax, constraint1, constraint2, constraint3,
0 <= {x1, x2, x3, x4, y1, y2, y3} <= 1}, {x1, x2, x3, x4, y1, y2,
y3}]
```

Hopefully the sample right here is evident. Is it doable to set this up in Mathematica to unravel for the restrict as $ntoinfty$? I’m largely having bother with the truth that I’ve so as to add a brand new constraint (and new variables) for will increase in $n$.

Many thanks,

Tom Waits