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

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 $$\beta \in C$$

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.