Find an optimal solution to the linear program and general


1. Consider the non-linear program P2 shown below:

          max -x2 + 4x - y2 + 12y

P2 =    s.t. -5x + 4y ≤ 20

           x2 - y ≤ 0

Starting at the solution (x, y) = (1, 4), run one iteration of the General Improving Search Algorithm to find a better solution. Use the gradient as the direction d.

2. Use the two-phase simplex algorithm to find an optimal solution to the linear program P2 (if such a solution exists).

           min -7X1 - 4X2 + 4X3

           S.t.  5X1 - 4X2 - 3x3 ≥ -3

P2 =            3x1 + 5x2 ≤ 6

                  3x2 2x3 ≥ 5

                  X1, X2, X3 ≥ 0

3. Use the two-phase simplex algorithm to find an optimal solution to the linear program P3 (if such a solution exists).

              max 4x1 + 5x2 - 3x3

              s.t. x1 + 2x2 + x3 = 10

P3 =        X1 - X2 ≥ 6

              X1 + 3X2 + X3 ≤ 14

              X1, X2, X3 ≥ 0

Given a feasible solution x to a linear program and an improving direction d if x + λd is feasible for all λ ≥ 0, then our optimization problem is unbounded.

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Operation Research: Find an optimal solution to the linear program and general
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