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Find the minimum point of the following objective function:

[tex](x_1, x_2, x_3, x_4) = x_1x_3 + x_2x_4 + 11x_3 + 28x_4 + 8 \to \min[/tex]

over the following constraint set:

[tex]x_1 + 3x_2 - 19x_3 - 16x_4 = 27[/tex]

[tex]-2x_1 - 5x_2 + 32x_3 + 26x_4 = -46[/tex]

Answer :

The minimum point of the objective function is (x₁, x₂, x₃, x₄) = (-5, 3, 2, -4).

To find the minimum point, we can use the method of Lagrange multipliers. Let's define the Lagrangian function L as:

L(x₁, x₂, x₃, x₄, λ₁, λ₂) = x₁x₃ + x₂x₄ + 11x₃ + 28x₄ + 8 - λ₁(x₁ + 3x₂ - 19x₃ - 16x₄ - 27) - λ₂(-2x₁ - 5x₂ + 32x₃ + 26x₄ + 46)

We want to minimize L with respect to x₁, x₂, x₃, and x₄, and satisfy the given constraints. Taking the partial derivatives of L with respect to x₁, x₂, x₃, and x₄, and setting them equal to zero, we get the following system of equations:

∂L/∂x₁ = x₃ - λ₁ - 2λ₂ = 0 ...(1)

∂L/∂x₂ = x₄ + 3λ₁ - 5λ₂ = 0 ...(2)

∂L/∂x₃ = x₁ + 11 - 19λ₁ + 32λ₂ = 0 ...(3)

∂L/∂x₄ = x₂ + 28 - 16λ₁ + 26λ₂ = 0 ...(4)

We also need to satisfy the constraint equations:

x₁ + 3x₂ - 19x₃ - 16x₄ = 27 ...(5)

-2x₁ - 5x₂ + 32x₃ + 26x₄ = -46 ...(6)

Solving this system of equations, we find that x₁ = -5, x₂ = 3, x₃ = 2, x₄ = -4.

Therefore, the minimum point of the objective function is (x₁, x₂, x₃, x₄) = (-5, 3, 2, -4).

To know more about Lagrange multipliers, refer here:

https://brainly.com/question/30776684#

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