0
$\begingroup$

In linear programming, if there are 'm' equations and 'n' variables, by making 'n-m' variables non basic(i.e. zero), we can get the corners of the feasible region. I could not understand how this works i.e., how we reach the corner points. Please explain.

Edited: I can understand that if the constraint coefficient matrix has full row rank ( m rows), keeping n-m variables zero will give a unique solution because 'm' independent vectors of 'm' dimensions combine to give a single vector of 'm' dimensions. But why should this unique solution (n dimension vector with at least n-m zero components) represent the corner point?

$\endgroup$
  • 1
    $\begingroup$ I suggest consulting your textbook or notes...surely this is covered somewhere. It's pretty fundamental to the theory of linear programming. $\endgroup$ – Math1000 Mar 8 at 3:34
  • $\begingroup$ @Math1000 Could you please suggest a book. My professor thought me just the simplex algorithm and no further explanation as to what happens in each step. $\endgroup$ – Mohan Mar 8 at 3:42
  • $\begingroup$ Is this related? math.stackexchange.com/questions/896388/… $\endgroup$ – David K Mar 8 at 4:44
  • $\begingroup$ See this thread for some recommended textbooks: math.stackexchange.com/questions/20643/linear-programming-books $\endgroup$ – Math1000 Mar 8 at 17:51
  • 1
    $\begingroup$ @Math1000 Thanks man, one of the books was really good. There was a really good explanation for why the basic feasible solution (unique solution with n-m zero components) is a corner of the polyheadron feasible region. $\endgroup$ – Mohan Mar 9 at 4:17
1
$\begingroup$

Guess you are talking about Simplex algorithm for solving linear programming problems? If you have more time, take a look at https://brilliant.org/wiki/linear-programming/, generally, it's based on the assumption that max/min function value must exist on corner points for linear programming problem setting, so we start from some point in the set to move up to the optimal by going through the corner points which help keep increasing/decreasing(depends on your optimization target, max/min) the function value. That procedure could be formalized in matrix expressions.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks for answering. I can understand the working of the algorithm. But why setting nonbasic variables to zero gives basic solution(i.e. corner points of the feasible region). $\endgroup$ – Mohan Mar 8 at 5:48
  • $\begingroup$ Kind of like (non-basic var)giving out the opportunity(to be 0) for maximizing/minimizing function value to those(basic var) who would do the work most efficiently(steepest direction according to gradient). $\endgroup$ – pmixer Mar 8 at 15:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.