A Survey on Pivot Rules for Linear Programming

3 : Abstract The purpose of this paper is to survey the various pivot rules of the simplex method or its variants that have been developed in the last two decades, starting from the appearance of the minimal index rule of Bland. We are mainly concerned with the niteness property of simplex type pivot rules. There are some other important topics in linear programming, e.g. complexity theory or implementations, that are not included in the scope of this paper. We do not discuss ellipsoid methods nor interior point methods. Well known classical results concerning the simplex method are also not particularly discussed in this survey, but the connection between the new methods and the classical ones are discussed if there is any. In this paper we discuss three classes of recently developed pivot rules for linear programming. The rst class (the largest one) of the pivot rules we discuss is the class of essentially combinatorial pivot rules. Namely these rules only use labeling and signs of the variables. These include minimal index type rules and recursive rules. The second class contains those pivot rules which can be considered in fact as variants or generalizations or specializations of Lemke's method, and so closely related to parametric programming. The last class of pivot rules discussed in this paper has the common feature that these rules all have close connections to certain interior point methods. Finally we mention some open problems for further study.