I have two variables:
$\kappa_y$ and $\kappa_x$
And three functions:
$M_y$=$M_y$($\kappa_y$, $\kappa_x$)
$M_x$=$M_x$($\kappa_y$, $\kappa_x$)
$F_z$=$F_z$($\kappa_y$, $\kappa_x$)
All these three functions are nonlinear and only way to evaluate each one for given $\kappa_y$, $\kappa_x$ is to do a intensive computation.
I want to find optimized values for both $\kappa_y$ and $\kappa_x$ in a way that it minimizes absolute value of each of these lines:
$F_z$($\kappa_y$, $\kappa_x$) - $F_t$
$C$ * $M_x$($\kappa_y$, $\kappa_x$) - $E$ * $M_y$($\kappa_y$, $\kappa_x$)
Note: Where $C$, $E$ and $F_t$ are constant values.
I always know a good starting point (say $\kappa_{y,0}$, $\kappa_{x,0}$) which are very very close to optimized values and if we assume gradient based methods can be used to find optimized values, what methods i can use for finding optimized values?
regarding gradient, changing $\kappa_y$ to $\kappa_y+d\kappa_y$ and $\kappa_x$ to $\kappa_x+d\kappa_x$ will result in:
$dF_z$ = $\frac {\partial F_z} {\partial \kappa_x} * d\kappa_x + \frac {\partial F_z} {\partial \kappa_y} * d\kappa_y $
$dM_y$ = $\frac {\partial M_y} {\partial \kappa_x} * d\kappa_x + \frac {\partial M_y} {\partial \kappa_y} * d\kappa_y $
$dM_x = \frac {\partial M_x} {\partial \kappa_x} * d\kappa_x + \frac {\partial M_x} {\partial \kappa_y} * d\kappa_y $
To minimize functions:
$F_z$ + $dF_z$ - $F_t$ = 0 (to minimize it) =>>
$dF_z$ = $F_t$ - $F_z$ = $\frac {\partial F_z} {\partial \kappa_x} * d\kappa_x + \frac {\partial F_z} {\partial \kappa_y} * d\kappa_y $
$C * (M_x + dM_x) + E * (M_y + dM_y) = 0$ (to minimize it) =>>
$ C.dM_x - E.dM_y = E.M_y - C.M_x$ =>>
$ C.(\frac {\partial M_x} {\partial \kappa_x} * d\kappa_x + \frac {\partial M_x} {\partial \kappa_y} * d\kappa_y ) - E.(\frac {\partial M_y} {\partial \kappa_x} * d\kappa_x + \frac {\partial M_y} {\partial \kappa_y} * d\kappa_y ) = E.M_y - C.M_x$ =>>
$d\kappa_x * (C * \frac {\partial M_x} {\partial \kappa_x} - E * \frac {\partial M_y} {\partial \kappa_x}) + d\kappa_y * (C * \frac {\partial M_x} {\partial \kappa_y} - E * \frac {\partial M_y} {\partial \kappa_y}) = E.M_y - C.M_x$ =>>
assuming: $I_1 = (C * \frac {\partial M_x} {\partial \kappa_x} - E * \frac {\partial M_y} {\partial \kappa_x})$ and $I_2 = (C * \frac {\partial M_x} {\partial \kappa_y} - E * \frac {\partial M_y} {\partial \kappa_y})$, $J_1 = \frac {\partial F_z} {\partial \kappa_x}$, $J_2 = \frac {\partial F_z} {\partial \kappa_y}$ will result in:
$d\kappa_x * I_1 + d\kappa_y * I_2 = E.M_y - C.M_x$
$d\kappa_x * J_1 + d\kappa_y * J_2 = F_t - F_z$
to matrix form:
$\begin{bmatrix}I_1 & I_2\\ J_1 & J_2\end{bmatrix} * \begin{bmatrix}d\kappa_x\\ d\kappa_y\end{bmatrix} = \begin{bmatrix}E.M_y - C.M_x\\ F_t - F_z\end{bmatrix}$
For using gradient based method this system should be solved each time for $d\kappa_x$ and $d\kappa_y$ until two target functions reach to 0 with desired tolerance.