Optimize $x^2+y^2+y-1$ subject to $x^2+y^2\leq1$

Question

I´ve been trying to solve this optimization problem using Lagrange methods. Analizing the interior of the domain, it is clear that it has a mínimum on $(0,-0.5)$. But analizing along the border, I found out the point $(0,1)$ using lagrange. The hessian matrix is positive definite so that point is a minimun too. But I´m not sure if what I did is correct. Is there another critical point?

Answer 1

With $f(x,y) = x^2+y^2+y-1$, follows the lagrangian

$$ L(x,y,\lambda,s) = f(x,y)+\lambda(x^2+y^2-1+s^2) $$

Here $s$ is a slack variable to transform the inequality into an equation. The stationary points are the solutions for

$$ \nabla L = \cases{2x+2\lambda x = 0\\ 2y+1+2\lambda y = 0\\ x^2+y^2-1+s^2=0\\ \lambda s = 0} $$

giving

$$ \left[ \begin{array}{ccccc} f& x & y & \lambda & s^2\\ -\frac{5}{4} & 0 & -\frac{1}{2} & 0 & \frac{3}{4} \\ -1 & 0 & -1 & -\frac{1}{2} & 0 \\ 1 & 0 & 1 & -\frac{3}{2} & 0 \\ \end{array} \right] $$

Follows a plot showing the level curves for $f(x,y)$ in black, the feasible region in light blue and the stationary points in red. Note that $s=0$ at the frontier points.