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  3. 2 we say a set c of points in r...

Question: 2 we say a set c of points in r...

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2. We say a set C of points in R is conver if for every pair x,y E C, all points on the line segment joining x and y are in C. Thus C is a convex set if for every pair x,y E C and any λ E (0.1) we have λχ + (1-λ)y E C. Note that λχ + (1-A)y = y + λ(x-y) which for λ E [0, 1], yields the line segment joining x and y. Let A be an mx n matrix and b a given vector in R. Show that C={x e Rn : Ax b, x 0) is a convex set. (That is, the domain of a LP is convex) 3. Consider the LP: Maximize 71 S 2 a) Solve the above LP using our Two Phase Method b) Write the Dual LP of the above Primal LP. Solve the Dual LP using our Two Phase Method. c) Verify that y,1 cision andェslack satisfy the magic coefficients. (That is, ydecision--the d) Do rdecision and yalack satisfy the magic coefficients? (That is, do decisionthe coefi 4. Show that the three inequalities You will need to convert the Dual LP into standard inequality form first coefficients of Tatack in the z-row of the final (primal) dictionary.) cients of yslack in the w-row of the final (dual) dictionary?) have no solution r, y, z with T, y,z 20 by using our two phase method. In addition, try to show infeasibility by finding a (positive) linear combination of the three inequalities which provides an obvious contradiction to x,y, z 2 0. The magic coefficients can be found in the final w row as the negative of the coefficients of the three slack variables you introduced.

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