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Polynomial-Time Interior-Point Methods

  • Yurii Nesterov
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 137)

Abstract

In this section, we present the problem classes and complexity bounds of polynomial-time interior-point methods. These methods are based on the notion of a self-concordant function. It appears that such a function can be easily minimized by the Newton’s Method. On the other hand, an important subclass of these functions, the self-concordant barriers, can be used in the framework of path-following schemes. Moreover, it can be proved that we can follow the corresponding central path with polynomial-time complexity. The size of the steps in the penalty coefficient of the central path depends on the corresponding barrier parameter. It appears that for any convex set there exists a self-concordant barrier with parameter proportional to the dimension of the space of variables. On the other hand, for any convex set with explicit structure, such a barrier with a reasonable value of parameter can be constructed by simple combination rules. We present applications of this technique to Linear and Quadratic Optimization, Linear Matrix Inequalities and other optimization problems.

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Yurii Nesterov
    • 1
  1. 1.CORE/INMACatholic University of LouvainLouvain-la-NeuveBelgium

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