Convexity and Optimization – Part III

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146 pages

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English

This book contains a brief description of general descent methods and a detailed study of Newton's method and the important class of so-called self-concordant functions.

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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.

- 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