Minimize a function regarding two coupling variables

Question

Given known matrics $A\in \mathbb R^{2\times 2}$ and known vectors $b\in \mathbb R^2, c\in \mathbb R^2$, for the two optimization variables $x\in \mathbb R^2$ and $y\in \mathbb R$, how to obtain the optimal variables $x$ and $y$ by minimizing the following expression: \begin{array}{rCl} x, y &=& \underset{x\in \mathbb R^2, y\in \mathbb R}{\operatorname{argmin}} x^\top A x - 2b^\top x - 2yc^\top x + 2y^2 +2 \end{array}

Is there an explicit solution for the two variables $x,y$? If no, how to obtain the optimal $x,y$?

Answer 1

The stationary conditions are

$$ \cases{ 2A x-2b-2y c=0\\ -2c^Tx+2y=0 } $$

and after substitution

$$ Ax-b-c^T x c=0 $$

developing we arrive at

$$ \cases{ a_{11}x_1+a_{12}x_2-b_1-(c_1x_1+c_2x_2)c_1=0\\ a_{21}x_1+a_{22}x_2-b_2-(c_1x_1+c_2x_2)c_2=0\\ } $$

or

$$ \cases{ (a_{11}-c_1^2)x_1+(a_{12}-c_1c_2)x_2-b_1=0\\ (a_{21}-c_1c_2)x_1+(a_{22}-c_2^2)x_2-b_2=0\\ } $$

or

$$ \left(A-c c^T\right)x = b $$

etc.