# Constraining regression coefficient to non-negative [duplicate]

I have a regression problem where I don't want the coefficients to be negative. Is setting negative coefficients of OLS to zero the same as constraining the coefficient to be non-zero and solving it through a convex optimiser as a quadratic problem?

I've thought about this from a geometric perspective. In OLS, $$\beta^TX$$ is the shadow of $$y$$ onto column space of $$X$$. The solution is where $$y - \beta^TX$$ is perpendicular to $$X$$. Thus, $$y - \beta^TX$$ is minimised. If one of the coefficients in $$\beta$$ is constrained (by replacing a negative value with zero, call this new vector $$\theta$$), $$y - \theta^TX$$ is still the vector with the smallest norm?

• This can be solved with quadratic optimization. Jan 13 at 6:56
• I fixed your title Jan 21 at 22:06
• Jan 21 at 22:23