2 - Foundations of Smooth Optimization

Summary

We outline theoretical foundations for smooth optimization problems. First, we define the different types of minimizers (solutions) of unconstrained optimization problems. Next, we state Taylor’s theorem, the fundamental theorem of smooth optimization, which allows us to approximate general smooth functions by simpler (linear or quadratic) functions based on information at the current point. We show how minima can be characterized by optimality conditions involving the gradient or Hessian, which can be checked in practice. Finally, we define the convexity of sets and functions, an important property that arises often in practice and that can be exploited by the algorithms described in the remainder of the book.