Reference no: EM13872259

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.