Descent and Interior-point Methods

Convexity and Optimization – Part III

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146 pages
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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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...

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