# Importance of the KKT Conditions

Does anyone know why the KKT Conditions are considered such a fundamental result in optimization?

As far as I understand, the KKT Conditions appear to be a set of conditions that if satisfied - suggest that the "constraints" within the optimization problem are not necessary in determining the solution?

Is this why the KKT Conditions are so important?

Thanks!

• In calculus, we learn that if $x^*$ is a minimizer for an unconstrained optimization problem then $\nabla f(x^*) = 0$. (Here $f:\mathbb R^n \to \mathbb R$ is the objective function that we're minimizing. I'm assuming $f$ is differentiable.) So we can solve unconstrained optimization problems by setting the derivative or gradient of $f$ equal to $0$ and solving for $x$. The analogous fact for constrained optimization problems is that a minimizer must satisfy the KKT conditions. The analogous strategy for solving constrained optimization problems is to solve the KKT conditions. Jan 13, 2022 at 4:27

• Might be worth noting that the equation $\nabla f(x^*) + \lambda^* \nabla g(x^*) = 0$ can be interpreted as saying that $x^*$ is a critical point of the Lagrangian $L(x,\lambda^*) = f(x) + \lambda^* g(x)$. If our optimization problem is convex, then $x^*$ is a minimizer of $L(x,\lambda^*)$. So if we magically know the value of $\lambda^*$, and if the function $x \mapsto L(x,\lambda^*)$ has a unique minimizer, then we can find $x^*$ by solving an unconstrained optimization problem. Jan 13, 2022 at 4:47