# Constrained Optimization

Ohh is this what they are doing with the Sinkhorn Divergence to solve the EMD??

They add a constraint to make it convex?

Learning this in the context of CS287.

**Original problem**: Find $max_{x}f(x)$ s.t. (subject to) $g(x)=0$

Original problem formulated as Lagrangian problem: Find $max_{x}min_{λ}L(x,λ)$, where $L(x,λ)=f(x)+λg(x)$.

Our objective is: $max_{x}min_{λ}(f(x)+λg(x))$ This Lagrangian equation respects the two conditions: $∂x∂L(x,λ) =0$ $∂λ∂L(x,λ) =0$ We can then solve it.

Why are these two equivalent?

Trying to convince myself that these two are equivalent. Pieter Abbeel was explaining that even in the the Lagrangian formulation, $g(x)$ is forced to be $0$ intuitively, else you will get punished.

I still don’t get it…