3-12 Consider the linear program -y + 5y2 -1 + y 3 1/2 = 2...

  

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3-12 Consider the linear program -y₁ + 5y2 -1 + y₂ ≤ 3 1/2 = 2 min s.t. Y/2 = 1 Yı, y2 = 0 at current solution y(¹) = (0,3). (a) List the condition for a direction Ay to be improving at y(¹), (b) Show that direction Ay = (1, -1) satis- fies your condition of part (a). (c) Determine which constraints are active at y(¹). ✔ (d) List and justify all conditions for any di- rection Ay to be feasible at point y(¹) (e) Show that direction Ay = (1, -1) sat- isfies your conditions of part (d), deter- mine the maximum feasible step À in that direction from y(¹), an compute the next solution point y(2) (f) Draw a 2-dimensional plot of the feasible space for this LP including contours of its objective. Then show how Ay = (1, -1) improves the objective, identify y(¹), and demonstrate how the same Ay preserves all constraints until it encounters an inac- tive one at the A of part (e) to produce y(2). 3-12 Consider the linear program -y₁ + 5y2 -1 + y₂ ≤ 3 1/2 = 2 min s.t. Y/2 = 1 Yı, y2 = 0 at current solution y(¹) = (0,3). (a) List the condition for a direction Ay to be improving at y(¹), (b) Show that direction Ay = (1, -1) satis- fies your condition of part (a). (c) Determine which constraints are active at y(¹). ✔ (d) List and justify all conditions for any di- rection Ay to be feasible at point y(¹) (e) Show that direction Ay = (1, -1) sat- isfies your conditions of part (d), deter- mine the maximum feasible step À in that direction from y(¹), an compute the next solution point y(2) (f) Draw a 2-dimensional plot of the feasible space for this LP including contours of its objective. Then show how Ay = (1, -1) improves the objective, identify y(¹), and demonstrate how the same Ay preserves all constraints until it encounters an inac- tive one at the A of part (e) to produce y(2).

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a The condition for a direction Deltay to be improving at y1 is that the directional derivative of t... View the full answer