Necessary Condition for Optimality

For a function $f:\mathbb{R} \rightarrow \bar{\mathbb{R}}$:

1. Assume $f$ is continuously differentiable over $\text{dom}(f)$, if $x^$ is a local minimizer of $f$ then $\nabla f(x^)=0$.
2. Assume $f$ is twice-continuously differentiable over $\text{dom}(f)$, if $x^$ is a local minimizer of $f$ then in addition to 1), it also holds that $\nabla^2f(x^) \succeq0$.

Notice that 1) is unidirectional — it is a necessary condition for optimality but not sufficient. This is because a saddle-point will also have a gradient of zero but will not be a local minimizer.

Sufficient Condition for Optimality

For a function $f:\mathbb{R} \rightarrow \bar{\mathbb{R}}$, assume $f$ is twice-continuously differentiable over $\text{dom}(f)$.

If $\nabla f(x^)=0$ and $\nabla^2 f(x^)\succ0$, then $x^*$ is a strict local minimizer.

This is a sufficient condition for optimality and it is bidirectional.