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Questions tagged [polyhedra]

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

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34 views

How to generate random bounded polytope by MATLAB defined by Ax=b, x≥0

How can one create a random bounded polytope in MATLAB, specified by the conditions $‎\lbrace‎x:~ Ax = b,~ x \geq 0‎\rbrace‎$
2 votes
1 answer
100 views

Maximum distance between two extreme points of a polyhedron

Let $P := \{x:Ax\le b\} $ and $V := \{x\in P: x = \frac{x^1+x^2}{2} \implies x^1 = x^2 = x\}$. i.e., $V$ is the set of extreme points of $P$. Assume $P$ is bounded. Is there a fast algorithm to ...
0 votes
0 answers
20 views

Supremum of a probabilistic function with ambiguity distribution set using Wasserstein metric

There is a proof of how to derive distributionally robust chance constraints with ellipsoid bound. $$\inf_{\mathbb{P}\in\mathcal{D}^{WD}} \mathbb{P}\{\|\mathbf{A\zeta-b}\|_2 \leq 1\} \geq 1-\epsilon$$ ...
4 votes
0 answers
95 views

Any recommendations for learning about polyhedra and integer programming?

My knowledge on convex polyhedra and systems of linear inequalities (facets, edges, Farkas Lemma, projections, duality, etc.) is very scattered, and I'l like to go through a book to solidify it. I'm ...
0 votes
2 answers
162 views

Need help with integer programming exercise

This is an exercise from Wolsey that I can't solve. Show how to go from Equivalence (1) to (2) and from Equivalence (2) to (3): $$ \begin{align} X &= \{ x \in \{0, 1\}^4~\mid~97x_1 + 32x_2 + ...
2 votes
0 answers
32 views

The facet-defining inequalities for a single resource scheduling problem

Suppose, there exists a scheduling problem $S$, in this case a single resource, with the following descriptions: $$ \text{conv(S)} = \{x \in \mathbb{R}^n \ | \ \forall \lambda_{i} \in \mathbb{R}^{n+}, ...
1 vote
1 answer
388 views

Discrete point inside a polygon formed by set of vertices

I am working on a problem where I have a set of 2D vertices and a test point. I want to chek whether the test point lies inside ...
2 votes
0 answers
60 views

System Stability constraints formulation

I am working with a system having a massless 2D plane and on that plane there is a rigid object with some mass placed on it. I want to support the plane with wooden sticks such that the system is ...
2 votes
1 answer
64 views

Does this kind of "partition" have a name?

Consider a convex polyhedron $A$. Assume we have subsets $A_1,\ldots,A_n$ of $A$ that are themselves covex polyhedra and are mutually disjoint except maybe sharing an edge, and that their union gives $...
6 votes
2 answers
972 views

Linear Program: Verify whether a feasible solution is an extreme point

My question is about a Linear Program (LP) of the form $\bf Ax\ge b$ with $\bf x\ge0$. From a theoretical standpoint: Given a feasible solution $\mathbf{x^{(0)}}$, how can we check (verify) whether it ...
2 votes
1 answer
113 views

Convex not strictly convex!

Update: Linear programming problems (LP) have a convex space, precisely vector space, such as a convex feasible region as pointed out by @prubin. Also, they may have either unique or multiple ...
5 votes
1 answer
189 views

How to enumerate all vertices of a polyhedron as a stream

I need to find the vertices of a polyhedron from the defining halfspaces. In a previous question libraries for enumerating all vertices were mentioned. However i have a very large polyhedron (~100,000 ...
2 votes
2 answers
114 views

Faces and Facets in a convex polyhedron

I am willing to find the number of faces and facets in a convex polyhedron. Suppose, in the cube polyhedron there exists $8$ vertices, $12$ edges, and $6$ faces. It satisfies the Euler equation as ...
0 votes
2 answers
204 views

On the non-sufficiency of total unimodularity of the constraint matrix in the definition of an integer polytope

Is there an example of an $m\times n$ integer matrix $A$ and an integer vector $b\in \mathbb {Z}^{m}$ such that the polyhedron $P := \{ x\in \mathbb {R}^{n} \mid A x \leq b\}$ is integer, while $A$ is ...
7 votes
1 answer
1k 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
1 answer
146 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 ...
1 vote
1 answer
174 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 ...
5 votes
1 answer
253 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 $\...
6 votes
0 answers
132 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
0 answers
79 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
0 answers
127 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?
9 votes
1 answer
293 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 ...
6 votes
1 answer
457 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.) ...
3 votes
0 answers
96 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 ...
13 votes
3 answers
7k 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 ...
5 votes
2 answers
183 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
0 answers
118 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
0 answers
72 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 ...
3 votes
1 answer
397 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 $n$, $m$ ...
10 votes
1 answer
350 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 ...
12 votes
1 answer
875 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 ...
10 votes
1 answer
201 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
3 answers
395 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
0 answers
86 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
6 answers
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
2 answers
2k 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?