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Descent and Interior-point Methods

Convexity and Optimization – Part III

by Lars-Åke Lindahl
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146
Language:  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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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
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About the Author

Lars-Åke Lindahl

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 retirement in 2010 and for more than 20 years served as chairman of the Math. Department.

He has given lectures in a variety of mathematical subjects such as Calculus, Linear Algebra, Fourier Analysis, Complex Analysis, Convex Optimization, Game Theory and Probability Theory, and he has also written several textbooks and compendia. After his retirement, he has been a consultant to Al Baha University, Saudi Arabia, with a mission to assist in the development of their master's program in Mathematics.

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