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Questions tagged [simplex]

For questions related to the simplex method for linear programming (LP), which solves LPs optimally by moving iteratively from in corner point of the feasible region to a better one.

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48 views

Solving convex separable programming problem using interior point method?

In my engineering application, all decision variables are non-negative and everything is convex separable. In addition to that, the only function that I am trying to approximate with grid point are $f(...
17 votes
4 answers
3k views

Where is the original Dantzig Simplex 1947 paper?

I see from here and many other sources that Dantzig invented the Simplex method in 1947. After much searching, I found that the earliest publication is this in 1956. Does anyone know where the ...
0 votes
0 answers
42 views

Proof of Dual Simplex Ratio Test

I have been trying to find proof of the ratio test for Dual Simplex, that is, selecting the variable that enters the base. Specifically, I was interested in knowing why by making that choice, reduced ...
2 votes
2 answers
507 views

Simplex method for multiple objectives

I am a user of Google OR Tools, which can interface with many LP & MIP solvers, plus it's own SAT based constraint programming solver. My question, in the context of OR-Tools, is: how should I ...
4 votes
3 answers
191 views

warmstarting simplex algorithm- how much can problems differ from each other?

I'm working on an implementation of the simplex algorithm. I want to solve problems in real time every 30 minutes. They could be interpreted as a classic transportation problem. I couldn't really say ...
2 votes
0 answers
211 views

Simplex Method Manual pivoting in GLPK for a Warm Start

My question is kind of related to this 2015 post at Stackoverflow. About Simplex method tableau pivoting in Linear Programming -- this is proving to be difficult in GLPK.. I am using Linux, GLPK 5.0 ...
8 votes
2 answers
1k views

Linear optimization problem with user-defined cost function

I have a linear optimization problem for which I am looking for a suitable optimization solution that can fulfill my requirements. Here is an explanation of the optimization problem: There are a ...
1 vote
3 answers
86 views

The dual values and change in the variables values

For a constraint as Ax <= b, the dual shows the change in the objective function if the RHS increases by 1 unit. Now my question is that how we can determine how the optimal values will change by 1 ...
7 votes
1 answer
1k views

Negative reduced cost for basic variable

I am observing something unusual : after solving a linear program, some basic variables have negative reduced costs (instead of $0$) : ...
2 votes
4 answers
140 views

Any good write up for Dual simplex with boxed variables?

I was wondering if there is any write up that shows how to perform dual simplex with boxed variables where l <= x <= u and preferably with a small example (...
0 votes
0 answers
90 views

why the -1 in basis column for simplex tableau?

When I look at the tableau generated by Gurobi for a simple 3D problem with five constraints, I get this: I expected to not have a -1 in basis columns 4, 5, 6 (referring to slack variables 1, 2, 3 if ...
2 votes
2 answers
572 views

How to handle strict inequalities?

Perhaps two trivial questions: What kind of problem is the following (please note the strict inequality)? How do we solve it? $$\min_{x\geq 0}\{c^\top x: Ax < b\}$$
0 votes
0 answers
69 views

Simplex algorithm for stochastic constraints?

The OR-Notes by J E Beasley states: Hence the problem: minimise 5x+6y subject to: Prob(a1x + a2y >= 3) >= 1-alpha x,y >= 0 ...
3 votes
0 answers
68 views

Block Simplex Algorithm, i.e., Block Active Set for Linear Programming

What investigation has there been of Block Simplex Algorithms, i.e., block active set for Linear Programming, i.e., block pivoting? This is a follow-up to Why do active set methods or the simplex ...
6 votes
1 answer
801 views

Network flow model - How can I turn this diagram into a matrix that when converted to RREF solves for max flow?

I have the following network flow model diagram and I have already calculated maximum flow using the R package igraph to be 28. However, what I would like to know ...
1 vote
0 answers
53 views

explain Givens rotation chain for maintaining Cholesky factorization

I'm attempting to implement the dual face algorithm from Pan's book chapter 22 (https://link.springer.com/book/10.1007/978-981-19-0147-8). The part in question is pasted here: Can you please explain ...
6 votes
1 answer
6k views

Interpretation of Reduced Costs

I am looking for an answer to a question I can't quite get behind. (continuation of Linear Programming: Integer and non-integer decision variables) I am given the following mathematical optimization ...
5 votes
2 answers
2k views

Simplex Multiplier

I am reading through a book which provides an example of a linear program given by \begin{align}\min&\quad-24y_{1}-28y_{2}\\\text{s.t.}&\quad6y_{1}+10y_{2} \leq 2400\\&\quad8y_{1}+5y_{2} \...
5 votes
2 answers
913 views

Why is it called the "Simplex" Algorithm/Method?

I have been trying to learn more about the Simplex Algorithm/Method. In particular, I am interested in knowing why this algorithm is called the "Simplex Algorithm". For instance, when ...
1 vote
0 answers
381 views

How to make unconstrained variables non-negative (as in excel solver) in AMPL?

This is a sequencing problem. I've got this variables ...
8 votes
1 answer
274 views

Why is the tailing off effect only a problem in column generation?

Why is the tailing off effect only a problem in column generation? If all columns were pregenerated, and one used the simplex method, wouldn't one see the tailing off effect? Is it simply not an issue ...
3 votes
1 answer
428 views

Are there two types of 'slack variables'?

In simplex algorithm, in order to handle inequation constraints, we need to convert them into equations by introducing so-called 'slack variables', like $$ \mathbf{a}^T\mathbf{x} + b \leq 0\quad\...
5 votes
1 answer
225 views

Short cuts when using the simplex algorithm

I haven't really kept myself up to date with research on the simplex algorithm for several years. I have taught linear programming, but it has not focused on the cutting edge of things. I remember ...
1 vote
1 answer
51 views

Computing simplex tableu for a given basis

I found the following problem in my book. I know that I can compute the simplex tableau , let's call it S for a basis X_b=(x_1, x2, x_5)^T as ...
4 votes
1 answer
447 views

Help me reproduce this tableau from the 'Integer Programming' book

From the Integer Programming book by Conforti et al, I've sniped the image below. At the bottom of this image there is the remnants of a tableau, presumably from several iterations of the simplex ...
3 votes
3 answers
345 views

Question regarding primal Simplex method

Given the following degenerate optimization problem \begin{align}\min&\quad c^Tx\\\text{s.t.}&\quad Ax=b,\\&\quad x\ge 0\end{align} Using primal simplex algorithm (either revised or ...
17 votes
3 answers
3k views

How Close Is Linear Programming Class to What Solvers Actually Implement for Pivot Algorithms

As part of a final project for my linear programming course, I have been asked to discuss implementations of pivot algorithms, including which combinations of the ideas we have talked about in class ...
1 vote
0 answers
104 views

How to incorporate artificial variables into the revised simplex method

Seems like we can convert >= constraints to <=, either by multiplying by -1 or using variable substitution. But for ...
4 votes
1 answer
494 views

Finding a dual feasible basis for use with the dual simplex algorithm

I have learnt that the dual simplex method requires reduced costs to be non-negative or else it cannot be used. I wanted to know what could be done to find a dual feasible basis and came across this ...
2 votes
1 answer
358 views

Inconsistent teachings on how to choose a non basic variable to enter the basis (primal simplex)

During the primal simplex algorithm, a non-basic variable must be chosen to enter the basis. Many resources on the subject choose a variable based solely on its coefficient in the row of the tableau ...
8 votes
1 answer
274 views

Column generation: decreasing value of restricted master problem

I am using column generation to solve a minimization problem. At a given iteration, my subproblem finds a column with reduced cost $-1$, and in the following restricted master problem, this new column ...
2 votes
1 answer
115 views

SoPlex basis file format

I'm running SoPlex from the command line to solve some linear programming problems. I'd like to get the basis of the solution, so I use the --writebas flag. However,...
4 votes
2 answers
283 views

Phase I of the simplex method and Farkas certificates

Phase I of the simplex method solves an auxiliary optimization problem to determine an initial basic feasible solution, or concludes that no such exists. Is there a way to use the solution of this ...
4 votes
1 answer
77 views

Upper bound on number of pivots to escape a degenerate point

Is it always possible to escape a degenerate point by a single pivot, or is it possible that several pivots are required? In other words is it possible to get away from a degenerate point by a single ...
7 votes
1 answer
292 views

Why are several of the decision variables zero at the corner point of a polytope?

I have the following equational Linear Program: \begin{align}\max&\quad c^T x\\\text{s.t.}&\quad Ax=b\\&\quad x\ge0\end{align} The matrix $A$ is $m\times n$, where $m\le n$, $c\in \mathbb{...
6 votes
0 answers
110 views

Dual instability, degeneracy, tailing off effect - Which are the causes and which are the effects?

Dual instability, degeneracy, and the tailing off effect are often mentioned together in papers. However, I cannot seem to find a clear explanation on which of these cause the other and vice versa? ...
7 votes
3 answers
384 views

Can the primal Simplex Method walk all optima in linear time?

It's typical that the Simplex Method implementation exits once it finds an optimum value. However, if I want to find all optima that exist at extreme points (not those that exist along a face), is it ...
6 votes
1 answer
1k views

Linear Programming: Integer and non-integer decision variables

I am looking for an answer to a question I can't quite get behind. I am given the following mathematical optimization problem: \begin{align}\min&\quad\sum_{t\in T}s_t\cdot z_t+h_t\cdot i_t+p_t\...
4 votes
1 answer
1k views

Best way to add dummy to transportation problem? Zero cost will be always chosen first?

I know that an unbalanced transportation problem could be made a balanced transportation problem by adding a dummy node which equals the difference between demand and supply. In literature, dummy ...
3 votes
2 answers
98 views

Transformations that leave the linear program unchanged

A typical linear program is written as $$L_0:\min_{x \geq 0; A^\top x \leq b}c^\top x.$$ Here, $x \in \mathbb{R}^n$, $c \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, and $b \in \mathbb{R}^m$. ...
2 votes
0 answers
162 views

Simplex - Network flow problem : Arc from 1 to P with infinite capacity

The Network - Maximum flow problem below aims to find the maximum flow using simplex method : With the LP as follow : LP : \begin{Bmatrix} Z(Max) = \sum_{i=1}^{m} fi \\ Af =0 \end{Bmatrix} ...
3 votes
1 answer
1k views

Simplex (GLPK) doesn't find a feasible solution on this simple assignment problem, but there is an obvious one

Problem Assign 11 projects to 11 students, based on their preference. For this example, each students chooses only one project, for simplicity shake (as shown below). Student 1 one chooses project 1, ...
2 votes
1 answer
145 views

Specific usecase of two-phase simplex algorithm

The problem below aims to find to most optimal way to transport the fuel : A company Er must transport a type of fuel from its two refineries Ra and Rb to its two points of sale PV1 and PV2. The ...
6 votes
0 answers
233 views

Provide basic solution to CLP

I'm using Pyomo to formulate an LP with approx 500,000 constraints and 200,000 decision variables. The LP is solved using CLP. Some instances fail to return even a feasible solution after many ...
6 votes
2 answers
972 views

Simplex algorithm and extreme points

For this question my short-hand is LP = linear program, BFS = basic feasible solution, SEF = standard equality form. Since the Simplex algorithm iterates from extreme point to extreme point (...
6 votes
0 answers
99 views

Proof that the leaving variable cannot be selected as the entering one in the next round

Using the Dantzig's pivoting rule, can it be proven that the leaving variable of one round cannot be selected as the entering variable in the next round?
4 votes
0 answers
267 views

Does anyone have the criss cross algorithm programming code to solve linear programming problems?

I have a project that requires programming code for the simplex algorithm and criss-cross algorithm. The purpose of this project is to compare the two methods. I've tried to find it, but the ...
3 votes
0 answers
440 views

On solving the Restricted Master Problem in Column Generation technique

I am working on developing a column generation (CG) based optimization framework for a large-scale airline crew pairing problem (a set-covering problem). First, I generate an initial feasible solution ...
3 votes
1 answer
279 views

Manually indicate initial basis for coin-or lp solver CLP

I have a set partitioning formulation with each constraint being an equality constraint to meet the given demand (right-hand side of the constraint). For each constraint, I have a slack and a surplus ...
7 votes
2 answers
274 views

Is the iteration-limited Simplex dual solution of a MIP node useful?

Idea Sometimes I encounter problems where Simplex spends many iterations for final convergence to the optimal objective value. Let's suppose, this happens when solving branch and bound-tree nodes as ...