The use of Grossone in Mathematical Programming and Operations Research

The concepts of infinity and infinitesimal in mathematics date back to anciens Greek and have always attracted great attention. Very recently, a new methodology has been proposed by Sergeyev for performing calculations with infinite and infinitesimal quantities, by introducing an infinite unit of measure expressed by the numeral grossone. An important characteristic of this novel approach is its attention to numerical aspects. In this paper we will present some possible applications and use of grossone in Operations Research and Mathematical Programming. In particular, we will show how the use of grossone can be beneficial in anti--cycling procedure for the well-known simplex method for solving Linear Programming Problems and in defining exact differentiable Penalty Functions in Nonlinear Programming

The s-monotone index selection rules for pivot algorithms of linear programming

In this paper we introduce the concept of s-monotone index selection rule for linear programming problems. We show that several known anti-cycling pivot rules like the minimal index, Last-In–First-Out and the most-often-selected-variable pivot rules are s-monotone index selection rules. Furthermore, we show a possible way to define new s-monotone pivot rules. We prove that several known algorithms like the primal (dual) simplex, MBU-simplex algorithms and criss-cross algorithm with s-monotone pivot rules are finite methods. We implemented primal simplex and primal MBU-simplex algorithms, in MATLAB, using three s-monotone index selection rules, the minimal-index, the Last-In–First-Out (LIFO) and the Most-Often-Selected-Variable (MOSV) index selection rule. Numerical results demonstrate the viability of the above listed s-monotone index selection rules in the framework of pivot algorithms

Parametric linear programming and anti-cycling pivoting rules

Bibliography: p. 13.Support in part from the Systems Theory and Operations Research Division of the National Science Foundation. ECS-83/6224 Support in part by Presidential Young Investigator grant of the National Science Foundation. 8451517-ECSby T.L. Magnanti and J.B. Orlin

Parametric Linear Programming and Anti-Cycling Pivoting Rules

The traditional perturbution (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore,restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric linear programming, we develop anti-cycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables

Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upo