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Basic Feasible Solutions: An algebraic characterization of extreme points for LP's in ``standard form''

For LP's in ``standard form'' the previous characterization of extreme points as the solution of n linearly independent binding constraints, which is, furthermore, a feasible point, can become even more concise. Consider, for instance, the LP



with m technological constraints and n variables. In ``standard form'', it becomes:



where I is the tex2html_wrap_inline1907 identity matrix, and tex2html_wrap_inline1909 is the vector of slack variables. It is interesting to notice that for every binding constraint from the m+n constraints of the original formulation (Eq. 25), one of the m+n variables of the ``standard form'' formulation must be equal to 0. Specifically, if the i-th technological constraint is binding, the corresponding slack variable tex2html_wrap_inline1917 . Similarly, if one of the sign restriction constraints is binding, then the corresponding varialbe tex2html_wrap_inline1919 . Since an extreme point of the feasible region of formulation 25 requires n binding constraints for its definition, it follows that n from the n+m variables in the corresponding solution of the ``standard form'' formulation (Eq. 26) must be equal to zero. Furthermore, since this point is uniquely defined, the system of equations defined by the m technological constraints in the ``standard form'' formulation and the m remaining variables, must have a unique solution. In other words, the columns of the ``standard form'' formulation corresponding to these m variables must be linearly independent (and their determinant must have a nonzero value). Finally, since the extreme point considered belongs in the feasible region of the problem, it follows that the unique solution of the aforementioned system of m equations in the m nonzero variables must be nonnegative (to meet the sign restrictions required by ``standard form'').

The structure of the ``standard form'' solutions corresponding to extreme points of the original feasible region (i.e., that defined with respect to the primary LP variables tex2html_wrap_inline1881 ), described for the example above, actually applies to any other LP in ``standard form''. We formally characterize this structure through the definition of basic feasible solutions for LP's in ``standard form'' (taken from Winston, ``Introduction to Mathematical Programming'').


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Next: Example: Up: An algebraic characterization of Previous: LP's in ``standard form''

UAL Data
Fri Jun 20 15:03:05 CDT 1997