A Q about convex optimality criterion

Question

Hope to ask about p. 139 of S. Boyd's cvx book:

x is optimal iff x is in X (feasible set) and

And the book use the following pic to illustrate:

My Q is: why there is a negative sign '-' in front of the gradient of f0?

I 'guess' the answer is this points to the direction of lower value? However, f0(x) is the lowest value; there is no f0(x') lower than f0(x). What does this '-' sign mean?

Thanks!

Answer 1

$f_0(x)$ is only the lowest value inside the feasible region. There are lower values but they are not feasible.