1.1 The Feasible Set of an LP Recall the standard-form LP (original) (raw)
Related papers
1998
In this essay, we will \discover" the dual problem associated with an LP. We will see how to interpret the meanings of the dual decision variables in the context of the original problem, and we will present some theorems (\facts") about the relationship between the optimal primal and dual solutions that will lead us to the key ideas of the simplex method for solving LPs.
Some comments on a linear programming problem
2018
Besides the very known two exits of the Simplex Algorithm we consider two more cases when at least a solution exists and to decide whether or not the solution is unique. This situation occurred in a linear programming problem, on one hand applying the Simplex Algorithm and on the other hand using Matlab command {\it linprog}, that led to the case of unbounded solution set and its construction. Some necessary conditions on data are given so that the set of solutions to be boundedless.
Linear Programming: Foundations and Extensions
Journal of the Operational Research Society, 1998
The text for this book was fonnated in Times-Roman using AMS-fb.T!Y((which is a macro package for Leslie Lamport's fb.T!Y(, which itself is a macro package for Donald Knuth's T !Y(text fonnatting system) and converted to pdf fonnat using PDFLATEX. The figures were produced using MicroSoft's POWERPOINT and were incorporated into the text as pdf files with the macro package GRAPHICX.TEX.
The many facets of linear programming
Mathematical Programming, 2002
At the last Mathematical Programming Symposium in Lausanne, we celebrated the 50th anniversary of the simplex method. Here, we are at or close to several other anniversaries relating to linear programming: the sixtieth of Kantorovich's 1939 paper on "Mathematical Methods in the Organization and Planning of Production" (and the fortieth of its appearance in the Western literature) [55]; the fiftieth of the historic 0th Mathematical Programming Symposium which took place in Chicago in 1949 on Activity Analysis of Production and Allocation [64]; the forty-fifth of Frisch's suggestion of the logarithmic barrier function for linear programming [37]; the twenty-fifth of the awarding of the 1975 Nobel Copyright (C) by Springer-Verlag. Mathematical Programming, 91 (2002), 417-436.
2013
1≤i≤m yibi. Now let P = {x ∈ (R) | Ax ≤ b and x ≥ 0}. Suppose x is a feasible point for our primal linear programming problem, so x ∈ P , i.e., Ax ≤ b and x ≥ 0. Also suppose we can choose y1, . . . , ym defining an m-covector y such that (yA)i ≥ ci and yi ≥ 0 for i = 1, . . . , n. Then c x ≤ yAx, and we have yAx ≤ yb, so altogether we have cx ≤ yAx ≤ yb. We see that yAx and yb are both upper-bounds of our primal objective function ζ(x) = cx for admissible choices of x and y, and yb is independent of x, so for any admissible choice of y, yb is an upper-bound of cx for all feasible points x ∈ P .
An Artificial-Free Simplex-Type Algorithm for General LP Models
Mathematical and Computer Modelling, 1997
The simplex algorithm requires additional variables (artificial variables) for solving linear programs which lack feasibility at the origin point. Some students, however, particularly nonmathematics majors, have difficulty understanding the intuitive notion of artificial variables.A new general purpose solution algorithm obviates the use of artificial variables. The algorithm consists of two phases. Phase 1 searches for a feasible segment of the boundary hyper-plane (a face of feasible region or an intersection of several faces) by using rules similar to the ordinary simplex. Each successive iteration augments the basic variable set, BVS, by including another hyper-plane, until the BVS is full, which specifies a feasible vertex. In this phase, movements are on faces of the feasible region rather than from a vertex to a vertex. This phase terminates successfully (or indicates the infeasibility of the problem) with a finite number of iterations, which is at most equal to the number of constraints. The second phase uses exactly the ordinary simplex rules (if needed) to achieve optimality. This unification with the simplex method is achieved by augmenting the feasible BVS, which is always initially considered empty at the beginning of Phase 1. The algorithm working space is the space of the original (decision, slack and surplus) variables in the primal problem. It also provides a solution to the dual problem with useful information. Geometric interpretation of the strategic process with some illustrative numerical examples are also presented. For teaching purposes you may try: A tabular simplex-type algorithm as a teaching aid for general LP models H Arsham Mathematical and Computer Modelling 12(8):1051-1056, 1989 Available at: http://home.ubalt.edu/ntsbarsh/Push\_pull\_original.pdf
Pathways to the Optimal Set in Linear Programming
Progress in Mathematical Programming, 1989
This chapter presents continuous paths leading to the set of optimal solutions of a linear programming problem. These paths are derived from the weighted logarithmic barrier function. The defining equations are bilinear and have some nice primal-dual symmetry properties. Extensions to the general linear complementarity problem are indicated. This chapter was published in the Proceedings of the 7th Mathematical Programming Symposium of Japan, Nagoya, Japan, pp. 1-35, and is referenced by several other papers in this volume. Since those Proceedings are not highly available, it is reproduced here in its original form.
Associated polyhedra and dual linear programs
Beitrage Zur Algebra Und Geometrie, 2010
The duality theorem of linear programming is set in a very general context, which is then mediated through the context of associated polyhedra; these latter are related by the representation theory of polyhedra. A feature of this approach is that it is made evident that two complementarity conditions are involved in the theorem.