Minimizing a function (cost function) of 3 variables

Question

I have to minimize the following cost function : $$cost\_func(\alpha,\beta,\gamma)=\sum_{i=1}^{5} \left( z_i - \alpha - \beta e^{2x_i+y_i} - \gamma e^{4 x_i+2 y_i}\right)^2 $$ where $x_i,y_i,z_i, i=1,...,5$ are constants. I know that there are deterministic,non-deterministic and mixed methods but is there any way to explore which one I can apply to this problem ? Is there any idea what I can do?

Thanks in advance

Answer 1

With the simplification

$$w_i = e^{2x_i+y_i}$$

$$v_i = w_i^2 = e^{4x_i+2y_i}$$

$$t_i = w_i^3 = e^{6x_i+3y_i}$$

$$u_i = w_i^4 = e^{8x_i+4y_i}$$

the following highly regular system of linear equation has to be solved:

$$5\alpha+\beta\sum_{i=1}^5w_i+\gamma\sum_{i=1}^5v_i = \sum_{i=1}^5z_i$$

$$\alpha\sum_{i=1}^5w_i + \beta\sum_{i=1}^5v_i + \gamma\sum_{i=1}^5t_i = \sum_{i=1}^5w_i z_i$$

$$\alpha\sum_{i=1}^5v_i + \beta\sum_{i=1}^5t_i + \gamma\sum_{i=1}^5u_i = \sum_{i=1}^5v_i z_i$$

For the resulting $\alpha$, $\beta$ and $\gamma$, the gradient of the costfunction is zero.