Interior point methods, a decade after Karmarkar—a survey, with application to the smallest eigenvalue problem

Abstract

The introduction of Karmarkar's polynomial algorithm for linear programming (LP) in 1984 has influenced wide areas in the field of optimization. While in the 1980s emphasis was on developing and implementing efficient variants of interior point methods for LP, the 1990s have shown applicability to certain structured nonlinear programming and combinatorial problems. We will give a historical account of the developments and illustrate the typical results by analyzing a new method for computing the smallest eigenvalue of a matrix. We formulate this latter problem as a so‐called semidefinite optimization problem. Semidefinite optimization has recently gained much attention since it has a lot of applications in various fields (like control and system theory, combinatorial optimization, algebra, statistics, structural design) and semidefinite problems can be efficiently solved with interior point methods.