What does the notation in a max/min statement mean, in a generalized Lagrangian?

Question

I'm studying about Lagrange duality and I I have the following problem:

$$ \min_{w} f(w) $$ s.t. $$ h_i(w) = 0, i = 1,...,l $$ $$ g_i(w) \leq 0, i = 1,...,k $$

So, somewhere in the text says that I should consider this quantity:

$$ \theta_p(w) = \max_{\alpha,\beta:\alpha_i \geq 0} \mathcal{L}(w,\alpha,\beta) $$

My question is, how should I read and interpret that max statement in the latter formula? Is it max over alpha, beta where alpha and beta are positive, or just alpha is positive?

Answer 1

To expand on @Florian 's comment.

$\beta$ are unconstrained in the dual problem because they correspond to equality constraints in the primal (i.e., original) problem.

$\alpha$ are constrained to be non-negative in the dual problem because they correspond to $\le$ inequality constraints in the primal problem.