<i>...a convex function can just be thought of as a function with a global maximum/minimum and no other local maximum/minimum.</i><p>This is false. The convex function exp(x) has no global max or min. The function f(x,y)=x^2 has infinitely many global minima.<p>The nice thing about convex functions is that the set of places where they achieve a minimum is a convex set: the empty set for exp(x) and the line x=0 for f(x,y)=x^2.