Can the following be solved analytically?
minimize $$ \ V(x) = |x_1-2| + |x_2-2| \ \ ; \ \ [x_1,x_2] \in R^2$$
subject to:
$$ h_1(x) = x_1-x_2^2 \ge 0 $$
$$ h_2(x) = x_1^2+x_2^2-1 = 0 $$
I solved it numerically but I'm not really familiar with analytic solutions.
What I tried:
- substitution with $h_1(x)$ and $h_2(x)$
- a method using the lagrangian...didn't seem to give an explicit solution
As $x_1,x_2 \leq 1$ you have $V(x)=4-x_1-x_2$ on the constraint. Then according the lagrange multipliers you have a critical point \begin{align*} x_1^2+x_2^2=1\\ -1=\lambda x_1\\ -1=\lambda x_2\\ \end{align*} which has solution $x_1=x_2=\frac 1 {\sqrt 2}$ at which $V=4-\sqrt{2}$ . Compare this to the value at the endpoints of the constraint at $\left ( \frac{\sqrt{5}-1}{2}, \pm \sqrt{\frac{\sqrt{5}-1}{2}} \right )$ to see that the minimum is $2-\sqrt{2}$ at $(\frac 1 {\sqrt 2} , \frac 1 {\sqrt 2} )$ and the maximum is $\sqrt{\frac{\sqrt{5}-1}{2}}+\frac{9-\sqrt 5}{2}$ at $\left ( \frac{\sqrt{5}-1}{2}, - \sqrt{\frac{\sqrt{5}-1}{2}} \right )$