linear optimization of multiplication coefficients

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?

• Are the $c_n$ required to be nonnegative? – Rob Pratt Nov 20 '19 at 13:28
• @RobPratt Yes, good hint. I have edited my text. – Anne Bierhoff Nov 20 '19 at 13:47

1 Answer

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

• Thanks for your answer. I saw that ordinary least squares are a part of quadratic programming. Would ordinary least squares be a possible way to solve the problem? – Anne Bierhoff Nov 20 '19 at 18:14
• This is ordinary least squares with some side constraints on the regression coefficients. – Rob Pratt Nov 20 '19 at 18:26
• Currently, I have a hard time trying to translate the problem into a quadratic programming problem. So I created a new question: math.stackexchange.com/questions/3454546/… – Anne Bierhoff Nov 28 '19 at 13:58