## A question about the lagrangian $L(x,\lambda, \nu)$ in the dual function in Convex Optimization

Hi. My question is probably very simple to some of you that have experience in Convex Optimization. The dual function is defined as the infimum of the lagrangian $L(x,\lambda, \nu)$ over all $x\$ in the domain. The lagrangian is: $f_0(x)+\sum \lambda_i f_i(x)+\sum \nu_i h_i(x)$

My question is, if $x\$ is in the domain, it satisfies the equality constraints $h_i(x)$ and in that case, $h_i(x)=0$. So why do we even have to mention the equality constraints if they zero-out anyway?

Thanks a lot, I hope I wrote my question clearly.

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The domain in question is the intersection of all the domains of the functions $f_i,h_i$. Not all the points in the domain satisfy the conditions (such points constitute what's called the feasible set). Also keep in mind that the Lagrangian dual is often a relaxation of the original convex optimization and only gives you a lower bound, unless you have strong duality.