Consider the following system of inequalities:
$Ax=b$; $x\geq 0$;
A is a $m\times n$ (non-square) and sparse matrix in which some part of entries are rational. How this system can be solved without using linear programming?
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$\begingroup$ please explain what mean by "solved". $\endgroup$– Dima PasechnikCommented Jul 18, 2012 at 17:35
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$\begingroup$ I meant, how can the feasibility of this system be checked without using linear programming? $\endgroup$– StarCommented Jul 18, 2012 at 17:51
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$\begingroup$ There is no general way, but perhaps you can use one of the "theorems of the alternative" such as Farkas's Lemma, Gale's Theorem, etc. $\endgroup$– Yoav KallusCommented Jul 18, 2012 at 20:43
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$\begingroup$ Why do you want to avoid linear programming? $\endgroup$– GileadCommented Jul 18, 2012 at 20:45
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From http://www.faqs.org/faqs/linear-programming-faq/
Q6.4: "I just want to know whether or not a feasible solution exists."
A: From the standpoint of computational complexity, finding out if an LP model has a feasible solution is essentially as hard as actually finding the optimal LP solution, within a factor of 2 on average, in terms of effort in the Simplex Method; plug your problem into a normal LP solver with any objective function you like, such as c=0. For MIP models, it's also difficult - if there exists no feasible solution, then you must go through the entire Branch and Bound procedure (or whatever algorithm you use) to prove this. There are no shortcuts in general, unless you know something useful about your model's structure (e.g., if you are solving some form of a transportation problem, you may be able to assure feasibility by checking that the sources add up to at least as great a number as the sum of the destinations).
