Finding local minimum under constraint

Question

How to find the minimum of $f(x) = ||x-\mu||^2$, where $\mu = (1, 1)$ and $< x, \mu > = 0$ (the inner product is $0$)?

Answer 1

The problem is a convex optimization problem, with the constraint being $$h(x)=x^T\mu=0$$ The corresponding Lagrangian is $$L(x,\lambda)=(x-\mu)^T(x-\mu)+\lambda x^T\mu$$ By the first order necessary condition for minima $$\nabla L(x)=0\Rightarrow 2(x-\mu)+\lambda\mu=0\Rightarrow x=(1-\lambda/2)\mu$$ which from the constraint and from the fact that $\mu=(1,1)$ gives $\lambda=2$. This gives $x=0$. You can check that the second order sufficient condition for minima that is $\nabla^2 L(x)=2I$ is a positive definite matrix and hence $x=0$ is indeed a minimizer.

Answer 2

$$||x-\mu||^2 = \langle x-\mu, x- \mu\rangle = ||x||^2 + ||\mu||^2$$ since $\langle x, \mu\rangle =0$. The expression on the right is obviously minimal for $x=0$