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Problem 1 (10 pt.) Derive the dual problem of following linear programs using Lagrangian relaxation: (1) max s.t. (2) min s.t. 10x₁ - 4x2 +7x3 3x₁-x₂ + 2x3 = 25, x₁ - 2x2 + 3x3 ≤ 25, 5x₁ + x₂ + 2x3 ≥ 40, 2x1 - x2 + x3 ≤ 20, X1, X2 ≥ 0, X3 ≤0. 3x₁ + x₂ + x3 x₁2x₂ + x3 ≤ 11, - 4x₁ + x₂ + 2x3 ≥ 3, - 2x1 + x3 = 1, x₁ unrestricted, x₂, x3 ≥ 0.

Fig: 1