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I'm reading Dan Stefanica's book "A Linear Algebra Primer for Financial Engineering", which says in pp 92, $\S3.2$ that "...the Fundamental Thorem of Linear Programming, which, informally speacking, states that, if a set of linear inequalities implies another set of linear inequalities, then it does so trivially, i.e. by linear combinations."

However, wiki page says:"In applied mathematics, the fundamental theorem of linear programming, in a weak formulation, states that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners. Further, if an extreme value occurs at two corners, then it must also occur everywhere on the line segment between them."

Well, I'm a bit lost. Are these two the same thing? If yes, how to show that they are equivalent? If not, which theorem is Dan referring to?

athos
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  • if $Ax_1 = b$ and $Ax_2 = b$ then $A(\lambda x_1 + (1-\lambda)x_2) = b$. if $Ax_1 < b$ and $Ax_2 < b$ then $A(\lambda x_1 + (1-\lambda)x_2) < b$. this is your theorem 1, while the theorem 2 is the theorem underlying the simplex algorithm. these are not exactly the same, but can be deduced from each other ? – reuns Jul 22 '15 at 08:24
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    and you should not read financial engeneering books, maths are not capitalistics. – reuns Jul 22 '15 at 08:29
  • @reuns. Your last comment is very good ! – Claude Leibovici Jul 22 '15 at 08:45
  • @reuns let's say i got to make a living... – athos Jul 22 '15 at 13:10

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