### Highlights fromCON2VERT - constraints to vertices

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# CON2VERT - constraints to vertices

22 Jun 2005 (Updated 11 Jul 2005)

Convert convex constraint inequalities into a set of vertices; i.e., polygon "vertex enumeration."

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Description

CON2VERT - convert a convex set of constraint inequalities into the set of vertices at the intersections of those inequalities;i.e., solve the "vertex enumeration" problem.

V = con2vert(A,b)

Converts the polytope (convex polygon, polyhedron, etc.) defined by the system of inequalities A*x <= b into a list of vertices V. Each ROW of V is a vertex. For n variables:
A = m x n matrix, where m >= n (m constraints, n variables)
b = m x 1 vector (m constraints)
V = p x n matrix (p vertices, n variables)

NOTES:
(1) This program emplyes a primal-dual polytope method.
(2) In dimensions higher than 2, duplicate vertices can appear using this method. This program detects duplicates at up to twelve digits of precision, then returns the unique vertices.
(3) Non-bounding constraints give erroneous results; therefore, the program detects non-bounding constraints and returns an error. You may wish to implement large "box" constraints on your variables if you need to induce bounding. For example, if x is a person's height in feet, the box constraint
-1 <= x <= 1000 would be a reasonable choice to induce boundedness, since no possible solution for x would be prohibited by the bounding box.
(4) This program requires that the feasible region have some finite extent in all dimensions. For example, the feasible region cannot be a line segment in 2-D space, or a plane in 3-D space.
(5) At least two dimensions are required.
(6) See companion function VERT2CON.
(7) Numerous examples are provided.
(8) ver 1.0: initial version, June 2005
(9) ver 1.1: enhanced redundancy checks, July 2005
(10) Written by Michael Kleder

Acknowledgements

This file inspired Representing Polyhedral Convex Hulls By Vertices Or (In)Equalities.

MATLAB release MATLAB 7.0.1 (R14SP1)
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30 Mar 2012

Michael Kleder,

Thanks for the useful code which your wrote. I used your code for the part of my dissemination and I need to cite your work. Would you please send me the information that I can cite your code? You can send your email to reza.jafari@okstate.edu
Regards,
Reza Jafari

18 Feb 2012

I want to believe in this tool, but problems with it continue to surface. I will change my rating if we at least get a reference explaining the algorithm that it uses.

As the latest difficulty, CON2VERT fails with this set of data,

A = [0 0 0 -1 2 -1; -1 2 -1 0 0 0;eye(6);-eye(6)];

b=[zeros(2,1); 10*ones(12,1)];

which looks like it should be fine (bounded and solid in R^6), but convhulln chokes on it in line 158.

Part of the problem is again that it cannot find an interior point c in line 151, but it doesn't end there. Even if I override the code and start at Line 158 with my own interior point

c=[ 2.5000 -5.0000 2.5000 1.4286 -4.2857 1.4286].'

I still get some infinite/nan vertices as output,

>> V(end,:)

ans =

NaN -Inf 10.0000 -10.0000 -10.0000 -10.0000

I suspect part of the problem is that Line 167 should be skipped if F is not full rank, but without a reference to the algorithm used, I can't be sure...

13 Feb 2012

Can you give the reference for the algorithm used

Thanks

06 Sep 2011

With a little more investigation into the problem, I've discovered that with the A,b data in my previous comment, the call to FMINSEARCH fails to find an initial point c that lies inside the polytope.

The test

if ef ~= 1...

fails to catch this. A better test would probably be

if f>0...

I don't really know what a good fix would be. From what I've read, finding an initial point inside a polytope is a sophisticated problem.

06 Sep 2011

Not sure if this file is being supported anymore, but there appears to be a problem. The point x=[2.5,3.5,1.8]'
is easily shown to satisfy
A*x<=b for the A,b data below. However, CON2VERT returns vertices that clearly do not enclose this point.

I'm wondering if the issue might be originating at this line

D = A ./ repmat(b,[1 size(A,2)]);

If any of the b(i) are negative, dividing through by b(i) will flip the direction of the corresponding inequality.

A=[1 0 0
0 1 0
0 0 1
-1 0 0
0 -1 0
0 0 -1
-0.387696185212551 -0.295276274520099 0.703237014010810
5.73326513862449 -0.422448628998633 -0.113292931747631
-3.79255233000727 -0.360645958414035 -0.673892379691217
-1.27188203250127 0.544750863219514 -0.791324376598499
-2.30540199426269 -2.50522755054822 -0.291771858420232
0.616578245027929 -0.654271120193835 0.574406987898971
-1.49508806389885 -2.68877782397954 -0.989225453798092
5.34556895341194 -0.717724903518731 0.589944082263178
-3.40485614479472 -0.0653696838939366 -1.37712939370203
-0.884185847288714 0.840027137739613 -1.49456139060931
-1.91770580905014 -2.20995127602812 -0.995008872431042
0.228882059815378 -0.949547394713933 1.27764400190978
-1.10739187868630 -2.39350154945944 -1.69246246780890
1.94071280861722 -0.783094587412668 -0.787185311438849
4.46138310612323 0.122302234220881 -0.904617308346131
3.42786314436180 -2.92767617954685 -0.405064790167863
5.11668689359657 0.231822491195202 -0.687699919646602
4.23817707472564 -3.11122645297817 -1.10251838554572
-2.52067029750601 -0.905396821633549 0.117431996907282
1.48715033574458 -2.14458159213419 0.382120521270985
-3.17597408497934 -1.01491707860787 -0.0994853917922465
2.29746426610842 -2.32813186556550 -0.315333074106875
-1.03351996176143 -3.04997841376774 0.499552518178267
-0.655303787473336 -0.109520256974320 -0.216917388699529
-0.223206031397585 -3.23352868719905 -0.197901077199593
-1.68882374923476 -3.15949867074206 0.282635129478739
-0.810313930363841 0.183550273431316 0.697453595377860
-0.878509818870922 -3.34304894417337 -0.414818465899121];
b=[6
4
2
0
0
0
-0.154147860211922
16.6671436561829
-8.83179250583593
-1.48693603688105
-8.92116339978305
0.423928365940654
-8.95704617639833
16.5129957959710
-8.67764464562401
-1.33278817666913
-8.76701553957113
0.269780505728732
-8.80289831618640
7.83535115034699
15.1802076193019
7.74598025639987
16.2432152902423
7.71009747978459
-7.34485646895489
-0.0893708939471196
-8.40786413989528
-0.125253670562397
-7.43422736290201
-1.06300767094039
-7.47011013951728
-8.49723503384240
0.0358827766152754
-8.53311781045768];

15 Feb 2011

Hi, is there a way to resolve the error:
"??? Error using ==> con2vert at 161
Non-bounding constraints detected. (Consider box constraints on variables.)" ?

31 Jul 2006

I wish the program could determinate which vertices belong to the same facet.

27 Jun 2006

legal

05 Apr 2006

Great! It helped me to do my asssignment!

14 Feb 2006

That's what I've been looking for.
No problems.

07 Feb 2006

works for me. No problems so far. good job! and thanks.