GATE 2020 MA – Question 28
The primal minimizes $x_1+2x_2$ subject to $4x_1-x_2\geq8$, $2x_1+x_2\geq10$, $-x_1+x_2\leq7$, $x_1,x_2\geq0$. Let x* and λ* be optimal primal and dual solutions. Find $x_1^*+x_2^*+\lambda_1^*+\lambda_2^*+\lambda_3^*$, to one decimal place.
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Correct answer: 5.5 to 5.5
Explanation
The second constraint implies $x_1+2x_2=\frac12(2x_1+x_2)+\frac32x_2\geq5$. The point (5,0) attains 5 and satisfies the other constraints, so it is optimal. Only the second primal constraint is active, and a valid optimal dual vector is (0,1/2,0), attaining dual value 10/2 = 5. Complementarity and the active positive x₁ give these multipliers. The requested sum is 5 + 0 + 0 + 0.5 + 0 = 5.5.