Questions tagged [polyhedra]

For questions on the set of points that satisfy a finite set of linear inequalities.

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6
votes
1answer
89 views

Extreme rays of a small polyhedral cone: How do I get them?

In a nutshell I have a small 2-dimensional polyhedral cone. $$C=\{(x_1,x_2): 2x_1-x_2 \leq 0, x_1+3x_2 \leq 0\}$$ I am looking for a simple, illustrative, procedure to get its extreme rays. Any ...
3
votes
1answer
87 views

Extreme points for capacitated flow polytope

Given a network as a directed acyclic graph with a source and sink, and non-negative edge capacities, I am interested in the extreme source-sink flows. More formally, I am interested in a ...
4
votes
1answer
121 views

Extreme points of a simple polyhedron

Consider the polyhedron given by the set of inequalities \begin{align} \mathbf{b}^T\mathbf{x} ~&\leq~ c \\ \mathbf{e}^T\mathbf{x} - 1 ~&\leq~0 \\ \mathbf{x}~&\geq~0 \end{align} where $\...
1
vote
1answer
159 views

References to publications on representation of any boolean function as a system of linear inequalities

It is known that any boolean function may be represented, in some sense, as a system of linear inequalities. But my rather intensive literature search brought a little references. I will appreciate ...
7
votes
0answers
93 views

Characterization for total dual integrality

A problem I study reduces to whether the polyhedron $P=\{\mathbf{x}\mid A\mathbf{x}=\mathbf{1}, \mathbf{x}\geq0\}$ is integral ($A$ is a matrix with coefficients in $\{0,1\}$). I know that the ...
1
vote
0answers
69 views

Decomposition of Polyhedra

There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$ \...
1
vote
0answers
47 views

Vertices of Polytope using Gurobi

Is there any way I can obtain all the vertices of a polytope using Gurobi? If this isn't possible, can I log all the intermediate vertices that Simplex finds before it hits the optimal one?
6
votes
1answer
212 views

Extreme point and extreme ray of a network flow problem

"It is a well-known result in network flow theory that an extreme point and an extreme ray of the polyhedron defined by the convex hull of feasible region corresponds to a path and cycle (resp.) ...
9
votes
2answers
162 views

How to get all the facet inequalities from a set of valid inequalities?

For a given set of valid inequalities $\cal V$ $$ \left\{\sum_{i}^n w_k x_i + c_k \le 0\right\}_k $$ we can obtain a polyhedron $P$ in $n$-dimensional space. It's known that the polyhedron $P$ can be ...
3
votes
0answers
71 views

Appropriate Rotation Matrix in Nonconvex Optimization with Barrier

Let $ x \in \mathbb{R}^n_+$ be a variable such that $\sum_{i=1}^n x_i = 1$. In other words, $x$ is in a probability simplex. I am working on barrier-like functions in nonconvex optimization over such ...
5
votes
2answers
131 views

What are examples of integral polytopes, where there exists an algorithm to write a given point as a convex combination of integral points?

Given a set of integer points $S$, one is often interested in finding $\operatorname{conv}(S)$ or characterizing certain cases, where $\operatorname{conv}(S)$ is described by few inequalities. ...
5
votes
0answers
79 views

Polyhedra to Simplex by using convex combination of vertices

Optimization problems over linear constraints (defining a convex polyhedron) can be written as optimization over a simplex in a higher dimension. Let $\mathcal{P}$ be a bounded polyhedron, and the ...
3
votes
0answers
64 views

Theoretical aspect of using extended formulation

If I can show a polyhedron Y is an extended formulation of polyhedron X and every extreme point in Y is integral, does that automatically imply the projection of Y onto the variable space of X gives ...
9
votes
3answers
1k views

How to find all vertices of a polyhedron

I have a convex polyhedron given by a set of linear inequalities, for example: $$ x_1 \geq 0,~~ x_2 \geq 0, ~~x_3\geq 0 \\ x_1+x_2\leq 1,~~ x_2+x_3\leq 1,~~ x_3+x_1\leq 1 $$ I want to list all the ...
3
votes
1answer
147 views

Estimation of the number of optimum vertices

Consider any linear programming model of $n$ variables and $m$ constraints which has multiple optimum solutions. If it is possible, I'd like to know the lower and upper limits (in terms of $n$, $m$ ...
9
votes
1answer
178 views

Solving convex programs defined by separation oracles?

General question: What software can solve convex programs defined by a separation oracle? The objective function is concave, and the feasible set is a polytope. By a separation oracle I mean that I ...
10
votes
1answer
173 views

Finding the linear functions defining a polyhedron through integer data?

Let's say I have a bunch of linear functions $f_1,\cdots,f_n$ in $k$ variables; then $f_1,\cdots, f_n\le0$ defines a polyhedron $P$ in the $k$-dimensional space. What I'm looking for is going the ...
10
votes
3answers
265 views

Is the "reverse search" algorithm of David Avis the state-of-the-art method for finding discrete solutions to a system of linear inequalities?

Is the "reverse search" algorithm of David Avis the state-of-the-art method for finding discrete solutions to a system of linear inequalities? If it is not, then what is? For $m$ inequalities in $d$ ...
6
votes
0answers
79 views

What are the top three applications (in terms of number of citations) of the "reverse search" algorithm of David Avis?

I can see that this algorithm is quite popular, and that one of the original papers now has 820 citations on Google Scholar. However, what are the most highly cited applications of it? If in Google ...
25
votes
6answers
2k views

How to compare two different formulations of a problem?

I somewhat know how to compare two MILP formulations of a problem that both use the same set of decision variables (as in the classical MTZ vs DFJ formulations of the TSP). I was wondering how two ...
14
votes
2answers
1k views

Polyhedra, Polyhedron, Polytopes and Polygon

About Polyhedra, Polyhedron, Polytopes and Polygon, what do they mean in the context of linear programming and what is the difference between them?
12
votes
1answer
315 views

Simplest way to eliminate redundant constraints due to a new cut

I have a polyhedral set for constraining $x$: \begin{align} \mathcal{P} = \{x \in \mathbb{R}^n_{+} : \ Dx \leq d \} \end{align} where $D \in \mathbb{R}^{m \times n}, d \in \mathbb{R}^m$. I find the ...