Chapter 1 Recent Developments in Interior- Point Methods

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semide nite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs. We review some core developments in the area.