Let \(\displaystyle y_1, \dots, y_n \in \mathbb{R}^d\) and \(\displaystyle v \in \mathbb{R}^d\) be given. I want to solve the following optimization problem:
Minimize \(\displaystyle J(x)=\sum_{k=1}^{n}{\|x-y_k\|^2}\) subject to \(\displaystyle h(x)=\|x-v\|^2-1=0\).
I have computed the gradients, which are \(\displaystyle \nabla J(x) = \sum_{k=1}^{n}{2(x-y_k)} = 2nx - 2n \bar{y}\), where \(\displaystyle \bar{y}=\frac{1}{n}\sum_{k=1}^{n}{y_k}\), and \(\displaystyle \nabla h(x)=2(x-v)\).
Using Lagrange multipliers, we obtain \(\displaystyle 2n(x-\bar{y}) = \nabla J(x) = \lambda \nabla h(x) = 2 \lambda (x-v)\).
Now, we can apply the constraint by multiplying \(\displaystyle (x-v)^T\) to both sides. We get \(\displaystyle \lambda = n (x-v)^T (x-\bar{y}) \).
I'm stuck here, because plugging \(\displaystyle \lambda\) into the first equation does not seem to help me. Can anyone help me?
Minimize \(\displaystyle J(x)=\sum_{k=1}^{n}{\|x-y_k\|^2}\) subject to \(\displaystyle h(x)=\|x-v\|^2-1=0\).
I have computed the gradients, which are \(\displaystyle \nabla J(x) = \sum_{k=1}^{n}{2(x-y_k)} = 2nx - 2n \bar{y}\), where \(\displaystyle \bar{y}=\frac{1}{n}\sum_{k=1}^{n}{y_k}\), and \(\displaystyle \nabla h(x)=2(x-v)\).
Using Lagrange multipliers, we obtain \(\displaystyle 2n(x-\bar{y}) = \nabla J(x) = \lambda \nabla h(x) = 2 \lambda (x-v)\).
Now, we can apply the constraint by multiplying \(\displaystyle (x-v)^T\) to both sides. We get \(\displaystyle \lambda = n (x-v)^T (x-\bar{y}) \).
I'm stuck here, because plugging \(\displaystyle \lambda\) into the first equation does not seem to help me. Can anyone help me?