Lagrange condition and second-order conditions

Question

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is understanding this question. Minimize $||x-x_0||^2$ subject to $||x||^2 = 9$ where $x_0= (1, \sqrt{3})^T$. Find all points that satisfy Lagrange and determine if minimizer using second-order conditions. How do I transform the objective function and constraints into something that looks like $x_1^2$.. etc.?

Answer 1

So you're essentially minimizing $((x_{1} - 1) + (x_{2} - \sqrt{3}))^{2}$ subject to the constraint, $x_{1}^{2} + x_{2}^{2} = 9$. You can try using the Lagrange Multiplier method:

$L(x_{1}, x_{2}, \lambda) = ((x_{1} - 1) + (x_{2} - \sqrt{3}))^{2} - \lambda(x_{1}^{2} + x_{2}^{2} - 9).$

Now take the partial derivatives with respect to $x_{1}, x_{2}, \lambda$ and set them to $0$. Those are your candidates to check.