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.



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


