Given a homogeneous linear equations AX=0 (X≥0), how to get the nonnegative basic solution set?
I think there is a mistake in the phrasing of the question. Since x is not restricted to be nonnegative the basis of the nullspace of A is not sign-restricted.
So what you have is a convex polyhedral cone. For those you need of course the extreme rays - these are the ones you are looking for. Here is a classic paper on the subject:
http://www.unc.edu/~erdavid/articles/56.pdf
But I should think that you find descriptions on this topic in any linear programming book.
As Michael Patriksson noted you need to have a study on linear programming. Linear programming section of the following book can be useful:
linear programming and network flows, 2nd/ed. by Mokhtar S. Bazara, John J. Jarvis and Hanif D. Sherali .
You can find an answer for your question by going through some materials on Linear Programming. Some standard books I can suggest are the following.
1. Linear Programming- G Hadley - Publisher: Addison.
2. Linear Programming and Network Flows- M S Bazaraa and J J Jarvis- Publisher: Wiley
3. Linear Programming - Kutta G Murty - Publisher: Wiley
4. Linear Programming: An Introduction With Applications - A. SUltan- Create Space Independent Publishing Platform.
5. Understanding and Using Linear Programming - J Matousek, Bernd Gärtner-Springer
Hope that these books will be helpful to you...
Yes, those constraints define a polyhedral cone, which has only one extreme point, namely the origin. To enumerate the extreme rays you could use a software package like PORTA or cdd.
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