China
Africa
Explore chapters and articles related to this topic
The Simplex Method
Published in Shashi Kant Mishra, Bhagwat Ram, Introduction to Linear Programming with MATLAB®, 2017
Shashi Kant Mishra, Bhagwat Ram
[Feasible Solution]A vector x∈Rn $ x \in \mathbb R ^{n} $ satisfying Ax=b $ Ax=b $ , where x≥0 $ x \ge 0 $ is called a “feasible solution”. A feasible solution that is also basic is called a “basic feasible solution”.
Linear Programming
Published in Michael W. Carter, Camille C. Price, Ghaith Rabadi, Operations Research, 2018
Michael W. Carter, Camille C. Price, Ghaith Rabadi
To describe the mechanics of the algorithm, we must specify how an initial feasible solution is obtained, how a transition is made to a better basic feasible solution, and how to recognize an optimal solution. From any basic feasible solution, we have the assurance that, if a better solution exists at all, then there is an adjacent solution that is better than the current one. This is the principle on which the Simplex method is based; thus, an optimal solution is accessible from any starting basic feasible solution.
Linear Programming
Published in Albert G. Holzman, Mathematical Programming, 2020
Sometimes a two-phase method is employed when problems have artificial variables. Simply stated, the purpose of the first phase is to drive all artificial variables to zero by considering only the artificial variables in the objective function. If a basic feasible solution is obtained in Phase 1, then the original objective function is used to obtain the optimum solution to the original problem. If the artificial variables cannot be driven to zero in Phase 1, then no solution exists to the original problem.
Improving a primal–dual simplex-type algorithm using interior point methods
Published in Optimization, 2018
T. Glavelis, N. Ploskas, N. Samaras
Linear Programming (LP) is a significant research area in the field of operations research. Linear programs can be found in almost every type of scientific and engineering applications. The first approach for solving linear programming problems (LPs) came from George B. Dantzig who set the fundamental principles. Dantzig proposed the simplex algorithm [1] that starts from a basic feasible solution and moves from one basic feasible solution to an adjacent one until an optimum solution is found. The vast literature of operations research contains an extensive family of simplex-type algorithms. Although the simplex algorithm is widely-used until today, there exist many real-life applications that its performance degrades due to a phenomenon called degeneracy. Simplex-type algorithms may stall at a degenerate vertex for many iterations before moving to another vertex. Degeneracy and redundancy are very common in real-life applications. Consequently, many different approaches were presented in order to overcome degeneracy [2]. Over the last decades, researchers focused on: (i) the reduction of the number of iterations that the simplex algorithm performs [3–5] (for a review, see [6]), (ii) the reduction of the computational work involved in each iteration [7–10] (for a review, see [11]), and (iii) new variants of the simplex algorithm [12–15].