Appendix B On Real Analysis · The--optimal--value of an optimization problem is a central concept. The introduction of this concept forces us to go into the foundations of the real numbers and to consider the com- pleteness property. · In the proof of the envelope theorem, we use the inverse function theorem. · One of the two types of good behavior of optimization problems is convexity, as we see in chapter four. We give some results on convexity, for example, we describe methods to verify whether a function is convex.