Pivot versus interior point methods: pros and cons

Linear optimization (LO) is the fundamental problem of mathematical optimization. It admits an enormous number of applications in economics, engineering, science and many other fields. The three most significant classes of algorithms for solving LO problems are: pivot, ellipsoid and interior point methods. Because ellipsoid methods are not efficient in practice we will concentrate on the computationally successful simplex and primal-dual interior point methods only, and summarize the pros and cons of these algorithm classes.