Find the saddle point of $F(x_1,x_2,x_3,y_1,y_2,y_3)=(x_1-2x_2+x_3)y_1+(2x_1-2x_3)y_2+(-x_1+x_2)y_3$

Question

Finding the saddle points of $F(x_1,x_2,x_3,y_1,y_2,y_3)=(x_1-2x_2+x_3)y_1+(2x_1-2x_3)y_2$+$(-x_1+x_2)y_3$ subject to the constraints $x_1+x_2+x_3=1, y_1+y_2+y_3=1$. Show that the saddle point is $x=(\frac{1/3}{1/3},\frac{1/3}{1/3},\frac{1/3}{1/3}),y=(\frac{2}{7},\frac{1}{7},\frac{4}{7})$.
I know how to find the saddle point of a function with two variable by using $\Delta=(f_{12})^2-f_{11}f_{22}$, then when $\Delta>0$ the point is a saddle point.
But for this question, by using the constraints, we can reduce the variables from 6 to 4, but still can not use the $\Delta$ formula, and this question requires not to use lagrange multiplier. Thanks.

Answer 1

At a saddle point, all of the partial derivatives are 0. This leads to a system of equations: \begin{align} x_1 -2x_2 +x_3=0,\tag{1}\\ 2x_1 -2x_3=0,\tag{2}\\ x_2-x_1=0,\tag{3}\\ y_1+2y_2-y_3=0, \tag{4}\\ -2y_1 +y_3=0,\tag{5}\\ y_1-2y_2=0. \tag{6} \end{align} Adding 1 to both sides of equation (4) leads to \begin{equation} 2 y_1 = 1-3 y_2. \tag{7} \end{equation}

Solving for $y_1$ from equation (6) and substituting in (7) leads to $y_2= \frac{1}{7}$ and consequently $y_1=2y_2=\frac{2}{7}$ and $y_3=2y_1=\frac{4}{7}$.

Similarly, we can subtract 1 from both sides of equation (1) and use the constraint for x, which leads to $x_2=\frac{1}{3}$ and consequently from (3), $x_1=x_2=\frac{1}{3}$. Finally, from (3), we have $x_3=x_1=\frac{1}{3}$.

EDIT: Computing the Hessian matrix yields: \begin{equation} H = \begin{pmatrix} 0 & 0 & 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & -2 & 0 & 1\\ 0 & 0 & 0 & 1 & 0 & 0\\ 1 & -2 & 1 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 0\\ -1 & 1 & 0 & 0 & 0 & 0 \end{pmatrix}. \end{equation} Diagonalizing gives us the characteristic polynomial \begin{equation} \lambda^6-12\lambda^4 +27\lambda^2 -4=0. \end{equation}

This polynomial has a $\lambda \rightarrow -\lambda$ symmetry. So if we can find one non-zero real solution, we automatically show that the Hessian is indefinite (There are both positive and negative solutions). Therefore the critical point above is indeed a saddle point.

EDIT 2:

In case you find it difficult to actually find the roots of the polynomial, substitute $\rho=\lambda^2$, this gives \begin{equation} \rho^3-12\rho^2 +27\rho -4=0. \end{equation}

Now, using the fact that complex roots come in pairs and that we can see that zero is not a solution, we can deduce that there is at least one real solution of the cubic $\rho_0$ and therefore two opposing sign solutions $\lambda_{+/-} = \pm \sqrt{|\rho_0|}$.