linear optimization of multiplication coefficients

Question

There are $n$ multiplication coefficients for which optimized values are searched. There is a restriction that the multiplication coefficients are not allowed to be negative and must have a sum of $1.0$:

$\sum_{i =0}^n c_i = 1.0$

The coefficients are multiplied by an individual vector and summed up to a new vector:

$\pmb v^{sum} = \sum_{i =0}^n c_i \cdot \pmb v_i$

Finally, the mean squared error between $\pmb v^{sum}$ and a vector $\pmb u$ is formed:

$mse = \frac{1}{m}\sum_{i = 0}^m (u_i - v^{sum}_i)^2$

So I want to find optimized $c_i$ values for which the $mse$ is minimized.

What's the best way to do that?

Answer 1

Geometrically, this is a projection onto the convex hull of the $\mathbf{v}_i$. You can solve it as a quadratic programming problem.