For a convex problem $\min_x f(x) + g(x) $, KKT point $x^$ satisfies $0 \in \nabla f(x^) + \partial g(x^*) $

Question

Apologies for a basic question.

For a convex problem $$\min_x f(x) + g(x) ,$$ where assuming $f(x)$ is differentiable with $m$-Lipschitz continuous gradient, while $g(x)$ is closed convex proper.

Then, KKT point $x^*$ satisfies $$0 \in \nabla f(x^*) + \partial g(x^*) .$$

My question is: Can we say $\nabla f(x^*) = 0$ at the KKT point? If yes, then it means, the subgradient $\partial g \in 0$ at the KKT Point? or do I miss any assumption?

score 1 · Accepted Answer · answered Jan 09 '20 at 11:57

1

No, you cannot say this. To give a simple example, consider the functions $f, g \colon \mathbb R \to \mathbb R$ defined via $f(x) = -g(x) = x$. Then, every $x^* \in \mathbb R$ is a minimizer with $f'(x^*) = 1$ and $\partial g(x^*) = \{-1\}$.

answered Jan 09 '20 at 11:57

gerw

31,359

Ok, thank you. Can any assumptions help to say that the gradient is zero? – learning Jan 09 '20 at 12:08
I cannot think of any reasonable condition. Of course, $f \equiv 0$ or $g \equiv 0$ works... – gerw Jan 09 '20 at 18:40
thank you for the clarification. – learning Jan 09 '20 at 20:03

For a convex problem $\min_x f(x) + g(x) $, KKT point $x^*$ satisfies $0 \in \nabla f(x^*) + \partial g(x^*) $

1 Answers1

For a convex problem $\min_x f(x) + g(x) $, KKT point $x^$ satisfies $0 \in \nabla f(x^) + \partial g(x^*) $