Complementary slackness with Lagrange Multipliers in Convex Optimization

Question

I was perusing the wiki article on the topic of this post

https://en.wikipedia.org/wiki/Convex_optimization

In particular the section on the Lagrange multipliers:

I would appreciate more insight into condition #3

λ1g1(x) = 0, ..., λmgm(x) = 0 (complementary slackness).

Why are all of the products required to be zero individually rather than say the sum of all products equalling zero?

Answer 1

Because those are the individual constraints that are imposed in the Lagrangian definition. Therefore, they are individually required to be equal to zero. If the sum of them is required to be zero only, then clearly this doesnt imply that the constraints are satisfied.