I'm looking at a typical maximization problem: $f(\Theta) = \sum z_i * ln \theta_i$.
That is subjected to following constraints: $\sum \theta_i = 1$ and $\theta_i \ge 0$.
Basically, the resources that I'm looking at tend to solve it using Lagrangian, but only by introducing 1 multiplier: $$L(\Theta, \lambda) = \sum z_i * ln \theta_i + \lambda(1 - \sum \theta_i)$$
I'm a little bit confused, why this constraint $\theta_i \ge 0$ is ignored. To be more precise, I'm struggling to prove that with following optimization problem:
$$L(\Theta, \lambda, B) = \sum z_i * ln \theta_i + \lambda(1 - \sum \theta_i) - \sum \beta_i * \theta_i$$
will have $\beta_i$ as zeros in optimal case.
I appreciate your help, guys.