# Questions tagged [linear-programming]

Questions on linear programming, the optimization of a linear function subject to linear constraints.

1,442 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
31 views

### Is there a good term for pairs of related variables in a system?

(Non-mathematician here. Sorry). Suppose you have a problem with lots of unknowns. The problem allows many solutions (possibly infinite). Certain pairs of unknowns (you don't know which ones) ...
189 views

### Linear programming: choosing entering variable

maximize 10𝑥1 + 12𝑥2 +12𝑥3 subject to 𝑥1 + 2𝑥2 + 2𝑥3 + 𝑥4= 20 2𝑥1 + 𝑥2 + 2𝑥3+𝑥5= 20 2𝑥1 + 2𝑥2 + 𝑥3 +𝑥6= 20 𝑥1, … , 𝑥6 ≥ 0 This is my first step for simplex tableau x1 x2 ...
185 views

235 views

### Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$\max~f(x)\quad \mbox{s.t.}~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can ...
925 views

### Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
369 views

### Modeling a lower-bound constraint on a euclidean distance in quadratic programming

I have been trying to model a certain problem into a mainly linear program, but with some quadratic constraints since I don't think that is something that can be avoided. I hope to solve it either ...
610 views

### Prove an artificial variable that leaves the basis will never return.

This is in the context of the Big M Method in the simplex algorithm in linear programming. Prove an artificial variable that leaves the basis will never return. I have no idea how to start this. ...
690 views

### mathematical model of an assignment/scheduling problem

I am solving a scheduling problem and I am able to abstract it into an assignment problem of assigning 45 machines to 42 jobs. the assignment problem was given as having 14 jobs, each with 3 tasks and ...
76 views

### Simple Linear Program Problem in Constrained Optimization

Here is a subproblem I am having difficulties with: $$d = \arg\min_x \ c^Tx$$ subject to $$x: \sum_{i=1}^{n} x_i = 0,\quad x_i \ge -b_i$$ for some $b \in \mathbb R^n$. So I'm looking for an ...
764 views

### Proof of Simplex Method, Adjacent CPF Solutions

I was looking at justification as to why the simplex method runs and the basic arguments seem to rely on the follow: i)The optimal solution occurs at some vertex of the feasible region (CPF points) ...
98 views

### What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)?

What is a closed chain (or circuit) that is used in solving a transportation problem (a special type of linear programming problem)? I'm having some problems with it. Please clarify it. I have posted ...
239 views

88 views

### Rayleigh Quotient variant?

If $A$ is a covariance matrix and I want to get $\max X^TAX$ where each value of $x$ is between -1 and 1. Is there a closed-form solution for this? I understand when $X^TX = 1$ this becomes the ...
203 views

### Check feasibility of a system of integer linear equations

I'm currently working on a very large integer linear programme which cannot be solved within any reasonable time. The initial set of linear equations S={Ax<=b) is feasible. I want to add more ...
321 views

### How to use the simplex method for linear programs?

I believe to be missing something important in the Simplex algorithm, because I can't get it to work. From what I gather, there are three steps per iteration, given a matrix for a linear program in ...
125 views

### How to reformulate this Set covering problem?

I am trying to solve the following implementation of the set covering problem of a crew rostering problem. Here constraint (19), meant to create a 12-hour break between the different shifts taken by ...
264 views

### Post-optimality analysis: Change in one of the constraints

Consider the LP: \begin{array}{rccc} \max& \quad -3x_1&-x_2& & \\ \text{s.t.}& \quad 2x_1&+x_2 &\leq 3 \\ & -x_1&+x_2 &\geq 1 \\ &&x_1,x_2 &\geq ...
28 views

### Determining maximum number of groups - maybe Linear Programming

Given a set D dogs, C cats, and B birds, for each dog d in D, there is a set c(d) which indicates the set of cats that dog d likes and a set b(d) birds that dog d likes. How do I find the maximum ...
146 views

### Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
1k views

### Get reduced costs from simplex tableau

This is probably a dumb question... but I'm trying to find how to calculate the reduced cost for a particular variable based on the information in a simplex tableau after I've minimized a linear ...
I have a question regarding using Big M constraints to solve the following problem: Given: $a, b \ge 0$ and integers. $$2a + 5b \le 17\\ a + b \le 5\\ 3a + 6b \le 20$$ For at least two of the ...