A Geometric Algorithm for Solving the General Linear Programming Problem
A Geometric Algorithm for Solving the General Linear Programming Problem
James Thurber
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U * (x) |p| K(x)f since p is the gradient direction for the face F (whether we . , 10 can move in this direction or not) . Thus P '7T = 7^-(P*u' :: '(x))+ -Mp. P) > -^(p'u'^x))* lLEl(p. U*(x)) M |w| |w| |w| |w| = ■, l? |g| r (P«tt*(x)) > ; g +e »P J (p. U*(x)) = p. U*(xK |£u'(x)+ep| 5|u (x) |+e|p| Thus p»(w/|w[) > p«u (x) which contradicts our definition of u (x) (since (w/|w[)£S ). Thus we must conclude that u*(x) = = p/IpI. Corollary 1 . For x€D-D w, there exists y€D such that y-x = Ap for so...me p and some X > 0. Furthermore for this y, §5. The geometric algorithm . We now give the algebraic formulation of the algorithm which we described geometrically in Section 3. So suppose we are given the constraints Ax > b and we wish to maximize L(x) = p*x . We further make the assumption that we are given w°£D . Of course this is a strong assumption, but in Section 7, we will illustrate (using this geometric algorithm) how one ob- tains such a w° (if it exists). We of course also know the di- rections p corresponding to the faces F.
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