Question

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.

Answer 1

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.