Essential Mathematics for Convex Optimization

In this chapter, we (a) demonstrate that every local minimizer of a convex function is its global minimizer, (b) show that the Fermat rule provides a necessary and sufficient condition for an interior point of the domain of a convex function to be a global minimizer of the function, provided the function is differentiable at the point, (c) introduce radial and normal cones and express in terms of these cones the necessary and sufficient condition for a point to be the minimizer of a convex function whenever the function is differentiable at the point, (d) introduce the symmetry principle and (e) provide basic information on maximizers of convex functions.