## Abstract

Major breakthroughs in mathematical programming are often related to linear programming (LP). First of all, the introduction of the simplex algorithm by Dantzig in 1947 [37] had both a theoretical and practical impact in the field, maybe even initiated it. As direct consequences we mention the development of LP and its extensions [39], network problems and algorithms [56], nonlinear programming (NLP) [146], decomposition schemes [41], complementarity theory [33, 148], stochastic programming [38], cutting plane methods for large integer programs [40], etc. The reader is referred to [149] for an overview of the early history of mathematical programming. When in 1979 Khacijan [130] showed that the ellipsoid algorithm applied to the LP problem runs in polynomial time, this was not only of importance for the complexity theory of LP, it also had important implications for the complexity theory of many combinatorial optimization problems, as shown by Grötschel et al. [92]. Unfortunately, the good complexity didn’t lead to good computational efficiency in practice, causing the method to become merely a theoretical tool. However, no single development since the introduction of the simplex method has influenced the field of mathematical programming to such an extent as did the 1984 paper by Karmarkar [126] which had (and still has) a great impact on both the theory *and* practice of mathematical programming. Describing a new polynomial time algorithm (called *projective scaling algorithm*) for LP with better complexity than the ellipsoid method and claiming it to be extremely efficient in practice, Karmarkar triggered a tremendous amount of research on what is now commonly called *interior point methods*. Hundreds of researchers all over the world went into the subject, over 2000 papers were written (see Kranich [144] for a bibliography). For an overview of the developments in the theory of interior point methods for LP the reader is referred to surveys by Gonzaga [87] and Den Hertog [100]; the computational state—of—the—art is described in Lustig et al. [157].

## Keywords

Interior Point Linear Programming Problem Linear Complementarity Problem Interior Point Method Central Path## Preview

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