I need to Optimize two equations simultaneously, Is there an algorithm to do that? For example I am working on the below equations: $F(x,y,z,t) = (xy - zt)^2$ where to optimize the difference between those two multiplications doesn't mean to optimize the xy and zt separately. for example, is xy equals to 6 and zt equals 6.1, so the absolute value is equals to 0.1, but xy and zt themselves are not optimized which are 6 and 6.1.
what I need is to optimize those equations simultaneously:
$F(x,y,z,t) = (xy - zt).^2$ and $F21(x,y) = xy$ and $F22(z,t) = zt;$
is that possible?
I was thinking to used Lagrange multipliers, and use constraints, but in Lagrange multipliers, it doens't mean the constraints are optimized also? is that right.
Is there any method or algorithm to make that optimization? Thanks in advance.
I believe you need to think about your problem a little more.
Before you even start on any kind of multiple objective optimization scheme, notice that the set of variables $(x,y)$ and $(z,t)$ are not mutually identifiable since they only appear in the form $xy$ and $zt$. Any values satisfying $xy=zt$ will be optimizers of your quadratic function.
Also, what does it mean to optimize the product $xy$ or $zt$? This function can take on any real value, so this doesn't mean much without clarification. If you are referring to adding constraints, such as $xy=2$, $zt=-3$, or $x+z=10$, for example, then these can make your problem better posed and the method of Lagrange multipliers can be used to solve these problems.