Consider the Following Nonlinear Programming Problem

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The Answer is The two stages involve making the decisions on the values of the two variables in order.At stage n,let the state s_n be the remaining slack in the constraint

The two stages involve making the decisions on the values of the two variables in order.At stage n,let the state s<sub>n</sub> be the remaining slack in the constraint ,so s<sub>1</sub> = 4 and s<sub>2</sub> = 4 - .For n = 2, .For n = 1, .To maximize (2x<sub>1</sub> + 4 - )over 0 ≤ x<sub>1</sub> ≤ 2,we take the first and second derivatives. (2x<sub>1</sub> + 4 - )= 2 - 2x<sub>1</sub> = 0,so x<sub>1</sub> = 1 is a critical point. (2x<sub>1</sub> + 4 - )= -2,so x<sub>1</sub> = 1 is indeed the maximum over 0 ≤ x<sub>1</sub> ≤ 2.Since x<sub>1</sub> = 1 leaves a slack of in the constraint,the overall optimal solution is = 1, ,with Z* = 5.

,so s₁ = 4 and s₂ = 4 -

.For n = 2,

.For n = 1,

.To maximize (2x₁ + 4 -

)over 0 ≤ x₁ ≤ 2,we take the first and second derivatives.

(2x₁ + 4 -

)= 2 - 2x₁ = 0,so x₁ = 1 is a critical point.

(2x₁ + 4 -

)= -2,so x₁ = 1 is indeed the maximum over 0 ≤ x₁ ≤ 2.Since x₁ = 1 leaves a slack of

in the constraint,the overall optimal solution is

= 1,

,with Z* = 5.

Consider the Following Nonlinear Programming Problem

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