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