Basic and non basic variables in linear programming

I dont understand what are Basic and non basic variables,why we are talking them specially, what they have got to do with the rank of the coefficient matrix and augmented matrix ,and some deal with the linearly independent set corresponding to the decision variables , and some finding the determinant of the coefficient matrix.

and if the solution is not feasible will it be still a basic solution?

Any kind of help is appreciated.

also kindly suggest where this concept is illustrated book/link Thank you

• What is the basic idea of Basic and Non-Basic variables,any reference with good examples is requested? – BAYMAX Jan 21 '16 at 1:45

So in a linear programming problem, you have what is geometrically some sort of multidimensional object (polyhedron) and what is algebraically a matrix, or system of equations.

So in a such a problem, you usually want to get the extrema of the object, because it is at the corner points of this multidimensional object that the objective function (what the equations represent and are modelling) is at a max or min. So if you look at the algebra, since you have a matrix/system of equations, there is an assignment of variables (from the equation standpoint) or a solution vector (from the matrix standpoint) that identifies/is the coordinates of the extrema. Now some of these variables you will set equal to zero, and some you will solve for by setting the equations equal to each other. The ones you set equal to zero are "non basic variables". Any one not set equal is a "basic" variable.

For more details, go to http://home.ubalt.edu/ntsbarsh/opre640a/partiv.htm and conrol-f basic and start reading from there. Linear programming is pretty cool, good luck!

linear programming simplex algorithm :

• the admissible set is a convex polyhedron (a "simplex").

• furthermore the linear objective function proves that one of the corners of that polyhedron must be (one of) the optimal solution(s).

• the convexity of the objective function allows us to jump from one corner to another by applying the gradient descent algorithm.

• the final idea is that a set of "basic variables" uniquely determinates one of the polyhedron corners.

• Friend , i am facing a lot of trouble in terminology of basic non basic variables ,how to identify basic and non basic variables in a lp , based on linear algebra, kindly suggest a good reference . – BAYMAX Jan 21 '16 at 1:47