How to understand the sufficient condition for global optimum for a constrained optimization probelm

Question

How to understand the sufficient condition for global optimum? From my understanding, the global optimum should be 0 instead of $\geq 0$, where does this come from?

Consider the constrained optimization problem

$$\underset{\beta \in R^p}{\min}f(\beta) \space \space \space s.t. \space\beta \in C$$

where $f$ is a convex objective function, and $C$ is a convex constraint set

when $f$ is differentiable then a sufficient condition for a vector $\beta^* \in C$ to be a global optimum is :

$$ \langle \triangledown f(\beta^*),\beta - \beta^* \rangle \geq 0 $$ for any $\beta \in C$

Answer 1

When you don't have constraints in the set or the optimum is strictly inside $C$, you should have $\nabla f(\beta^*)=0$ in order to satisfy for all other $\beta$. Having zero-gradient is also a sufficient condition but doesn't consider all possible situations, e.g. $f(\beta)=-3\beta, \beta\in[1,2]\rightarrow\beta^*=2$.

Assume the condition is satisfied. Since $f$ is convex, $f(\beta)\geq f(\beta^*)+\nabla f(\beta^*) (\beta-\beta^*)$. Since the second term is non-negative, it means $f(\beta)\geq f(\beta^*)$. So, if the condition is satisfied for some $\beta^*$, that $\beta^*$ is the optimum. It's a sufficient condition.