this paper is to present mathematical ideas and

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 finiteness 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 first 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 variab...

We devise a new simplex pivot rule which has interesting theoretical properties. Beginning with a basic feasible solution, and any nonbasic variable having a negative reduced cost, the pivot rule produces a sequence of pivots such that ultimately the originally chosen nonbasic variable enters the basis, and all reduced costs which were originally nonnegative remain nonnegative. The pivot rule thus monotonically builds up to a dual feasible, and hence optimal, basis. A surprising property of the pivot rule is that the pivot sequence results in intermediate bases which are neither primal nor dual feasible. We prove correctness of the procedure, give a geometric interpretation, and relate it to other pivoting rules for linear programming.

. Assuming that the reader is familiar with both the primal and dual simplex methods, Zionts' criss-cross method can easily be explained. ffl It can be initialized by any, possibly both primal and dual infeasible basis . If the basis is optimal, we are done. If the basis is not optimal , then there are some primal or dual infeasible variables. One might choose any of these. It is advised to choose once a primal and then a dual infeasible variable, if possible. ffl If the selected variable is dual infeasible, then it enters the basis and the leaving variable is chosen among the primal feasible variables in such a way that primal feasibility of the currently primal feasible variables is preserved. If no such basis exchange is possible another infeasible variable is selected. ffl If the selected variable is primal infeasible, then it leaves the basis and the entering variable is chosen among th

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