This third and final part of Convexity and Optimization discusses some optimization methods which, when carefully implemented, are efficient numerical optimization algorithms. We begin with a very brief general description of descent methods and then proceed to a detailed study of Newton's method. One chapter is devoted to self-concordant functions, and the convergence rate of Newton's method when applied to self-concordant functions is studied. We conclude by studying of the complexity of LP-problems.

1. Descent methods
   1. General principles
   2. The gradient descent method
2. Newton’s method
   1. Newton decrement and Newton direction
   2. Newton’s method
   3. Equality constraints
3. Self-concordant functions
   1. Self-concordant functions
   2. Closed self-concordant functions
   3. Basic inequalities for the local seminorm
   4. Minimization
   5. Newton’s method for self-concordant functions
4. The path-following method
   1. Barrier and central path
   2. Path-following methods
   3. The path-following method with self-concordant barrier
   4. Self-concordant barriers
   5. The path-following method
   6. LP problems
   7. Complexity