Everyone with some background in Mathematics knows how to solve a system of linear equalities, since it is the basic subject in Linear Algebra. In many practical problems, however, also inequalities play a role. For example, a budget usually may not be larger than some specified amount. In such situations one may end up with a system of linear relations that not only contains equalities but also inequalities. Solving such a system requires methods and theory that go beyond the standard Mathematical knowledge. Nevertheless the topic has a rich history and is tightly related to the important topic of Linear Optimization, where the object is to nd the optimal (minimal or maximal) value of a linear function subject to linear constraints on the variables; the constraints may be either equality or inequality constraints. Both from a theoretical and computational point of view both topics are equivalent. In this chapter we describe the ideas underlying a new class of solution methods...

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this article the motivation for desiring an "interior" path, the concept of the complexity of solving a linear programming problem, a brief history of the developments in the area, and the status of the subject as of this writing are discussed. More complete surveys are given in Gonzaga (1991a,1991b,1992), Goldfarb and Todd (1989), Roos and Terlaky (1997), Roos, Terlaky and Vial (1997), Terlaky (1996), Ye (1997), Wright (1996) and Wright (1998). Generalizations to nonlinear problems are briefly discussed as well. For thorough treatment of interior point algorithms on those areas, the reader is referred to den Hertog (1993), Nesterov and Nemirovskii (1993) and Saigal, Vandenberghe and Wolkowicz (1998).

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