Descent and Interior-point Methods

Convexity and Optimization – Part III
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146 pages
Språk:
 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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Om forfatteren

Lars-Åke Lindahl obtained his mathematical education at Uppsala University and Institut Mittag-Leffler and got a Ph.D. in Mathematics in 1971 with a thesis on Harmonic Analysis. Shortly thereafter he was employed as senior lecturer in Mathematics at Uppsala University, where he remained until his reti...

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Content

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