Development of Interior-Point Methods

During the last fifteen years we have witnessed an explosive development in the area of optimization theory due to the introduction and development of interior-point methods. This development has quickly led to the development of new and more efficient optimization codes. In this talk, the basic elements of interior-point methods for linear programming will be discussed as well as extensions to convex programming, complementary problems, and semidefinite programming. Interior-point methods are polynomial and effective algorithms based on Newton’s method. With their introduction, the classical distinction between linear programming methods, based on the simplex algorithm and methods of nonlinear programming has largely disappeared. Also, a brief overview of some implementation issues and some modern optimization codes based on interior-point methods will be presented. By now, there are no doubts that for large-scale linear programming problems these new optimization codes are very often more efficient than classical optimization codes based on the simplex method.