Tagged Questions

1
vote
1answer
46 views

Strict inequality in MILP

I have a problem with the following constraint. There are 2 variables $p \in [0,1] \subseteq \mathcal{R}$ $\sigma \in [0,1] \subseteq \mathcal{Z}$ The constraint over the variables is $c - p < ...
2
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1answer
44 views

Simplex on Linear Program with equations

My linear program instead of inequations also contains one equation. I do not understand how to handle this in every tutorial I searched the procedure is to add slack variables to convert the ...
0
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1answer
18 views

Compute the point of contraction of a bounded region in $\mathbb{R}^n$

Say we have a list of linear inequalities that define a bounded region in $\mathbb{R}^n$. These inequalities are: $a_1 \cdot x \ge c_1, \dots, a_k \cdot x \ge c_k$. Assume general position (i.e. it ...
0
votes
2answers
26 views

Checking the existence of a solution for a set of linear equality and ineaulity equations

I would like to know if there is a method to check the existence of the solution for a given set of linear equations composed with both equalities and inequalities? I'm not interested in the solution, ...
0
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0answers
48 views

Solving an overdetermined system of inequalities using null-space arguments

The solutions to a linear system of equations: $$A\cdot x = b$$ (where $x$ is a $(n\times 1)$ column vector, $b$ is a $(m\times 1)$ column vector and $A$ is $(m\times n)$ matrix) can all be ...
0
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0answers
50 views

linear equations with inequality constraints

The problem is, given a set of linear equations $Ax=b$ such that the system is under-determined, and a set of linear inequalities $Cx\geq 0$, find a solution for the system. Does anyone know a general ...
3
votes
1answer
119 views

Dimension of solution space for system of linear inequalities

Let's say I have a system of inequalities: $Ax \leq g$ for some $A \in \mathbb{R}^{4\times4}$, $x \in \mathbb{R}^4$, $g \in \mathbb{R}^4$, and $A$ is full rank. Here, the $\leq$ denotes element-wise ...
6
votes
1answer
385 views

Farkas Lemma proof

I am trying to prove the Farkas Lemma using the Fourier-Motzkin elimination algorithm. From Wikipedia: Let A be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the ...
3
votes
1answer
96 views

Minimal set of inequalities

I have a set of $m$ linear inequalities in $R^n$, of the form $$ A x \leq b $$ These are automatically generated from the specification of my problem. Many of them could be removed because they are ...
1
vote
1answer
205 views

Convex hull of sets defined by (in)equalities

If you define the convex hull of a set $X$ as the set of all convex combinations of elements of $X$, it becomes difficult to decide if a given element $w$ belongs or not to $conv(X)$ (You have to ...
1
vote
1answer
167 views

Does the triangle inequality suffice to prove all minimum results on sums of absolute values of affine functions?

The title says it all ... more formally : let $n \geq 1$, and let $a_1, a_2 , \ldots ,a_n$ be positive numbers, let $b_1, b_2 , \ldots ,b_n$ be real numbers. Consider for $x\in {\mathbb R}$, $$ ...
2
votes
2answers
170 views

How to calculate volume given by inequalities?

I need to find the volume of the 3d space that is given by the following conditions: \begin{array}{c} 0 < x_1 < 1\\ 0 < x_2 < 1\\ 0 < x_3 < 1\\ x_1 + x_2 + x_3 < a. ...
3
votes
1answer
225 views

Finding tight constraints on a linear inequality

I have $a^\intercal M b > 0$, where $\forall a_i > 0$, $\forall b_j > 0$, and M is known. I'd like to find a tight linear constraint on $b$ which is independent of $a$ (other than the ...
2
votes
1answer
365 views

Sufficient Conditions for a Bounded Feasible Region in the Linear Programming Problem

I am working on a problem where it would be nice to prove that the feasible region of a LP problem is bounded, but where it is not necessary to solve any particular problem. In particular, given an ...