Optimality condition for a convex, unconstrained problem

Consider the unconstrained optimization problem

 min_x : f_0(x) ,

where f_0 : mathbf{R}^n rightarrow mathbf{R} is convex and differentiable.

The optimality condition

  forall : y in mathbf{X} ~:~ nabla f_0(x) ^T (y-x) ge 0 .

where the feasible set mathbf{X} is now the whole space mathbf{R}^n, reduces to

 nabla f_0(x) = 0.

Example: