Parametric linear programming and anti-cycling pivoting rules (original) (raw)
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The simplex algorithm is a widely used method for solving a linear programming problem (LP) which is first presented by George B. Dantzig. One of the important steps of the simplex algorithm is applying an appropriate pivot rule, the rule to select the entering variable. An effective pivot rule can lead to the optimal solution of LP with the small number of iterations. In a minimization problem, Dantzig’s pivot rule selects an entering variable corresponding to the most negative reduced cost. The concept is to have the maximum improvement in the objective value per unit step of the entering variable. However, in some problems, Dantzig’s rule may visit a large number of extreme points before reaching the optimal solution. In this paper, we propose a pivot rule that could reduce the number of such iterations over the Dantzig’s pivot rule. The idea is to have the maximum improvement in the objective value function by trying to block a leaving variable that makes a little change in the ...
An Efficient Method for Pivoting Free Variables in Linear Programming: A Computational Approach
Commonly used simplex method to solve linear programming problem do not allow variables to be negative during solution process and suggest to break each free variable (variable allowed to be negative) into difference of two non-negative variables. This transformation significantly increases the number of variables as well as after this the problem leaves its original variable space. , thus making the geometry of problem (during solution process) difficult to handle and understand. In this paper, we developed a natural generalization of simplex pivots for free variables. Described approach is capable of handling any general linear programming in its original variable space. In our computational study, the primary results showed that the new method outperforms simplex method on general LPs.
Symmetric primal dual simplex pivoting decision strategy (spdspds) for linear programming
2014
The Symmetric Primal-Dual Simplex Pivoting Decision Strategy (spdspds) is a novel iterative algorithm to solve linear programming (LP) problems. Each iteration is based on a systematic selection and application of one among the newly identified set of four (or possibly six) distinct types of simplex pivots defined over a symmetric primal-dual pair of LP. The two (or possibly four) types of classical (standard) simplex pivots are the Primal Standard Pivot with positive (or zero) indicator, and the Dual Standard Pivot with negative (or zero) indicator. The two newly identified pivot types are: the Primal Tricky Pivot with positive indicator and the Dual Tricky Pivot with negative indicator. If more than one candidate pivot element/cell is of the same type, then a selection among them can be made based on a measure of goodness that is defined as the decrease in the infeasibility index of such cells. If further pivoting is not possible, then the tableau is checked for the terminal type ...
Max-out-in pivot rule with cycling prevention for the simplex method
ScienceAsia, 2014
A max-out-in pivot rule is designed to solve a linear programming (LP) problem with a non-zero righthand side vector. It identifies the maximum of the leaving basic variable before selecting the associated entering nonbasic variable. Our method guarantees convergence after a finite number of iterations. The improvement of our pivot rule over Bland's rule is illustrated by some cycling LP examples. In addition, we report computational results obtained from two sets of LP problems. Among 100 simulated LP problems, the max-out-in pivot rule is significantly better than Bland's rule and Dantzig's rule according to the Wilcoxon signed rank test. Based on these results, we conclude that our method is best suited for degenerate LP problems.
2020
The Symmetric Primal-Dual Simplex Pivot Decision Strategy (spdspds) is a novel iterative algorithm to solve linear programming (LP) problems. Here, a simplex pivoting operation is considered simply as an exchange between a basic (dependent) variable and a non-basic (independent) variable, in the Tucker's Compact Symmetric Tableau (CST) which is a unique symmetric representation common to both the primal as well as the dual of a linear programming problem in its standard canonical form. From this viewpoint, the classical simplex pivoting operation of Dantzig may be considered as a restricted special case. The proposed spdspds allows for a wider selection of simplex pivot elements, spanning the choice among the entire set of all nonzero elements of the coefficient matrix in the tableau. An infeasibility index associated with an LP is defined as the sum of the number of primal variables and the number of dual variables that are infeasible. A measure of goodness as a global effectiv...