Polynomial-Time Interior-Point Methods

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.