Discussion
Consider the non-linear programming problem
Max. \(Z = 10x_{1} - x_{2}^{2}\)
Subject to the constraints :
\(x_{1} + x_{2} \le 14\)
\(-x_{1} + x_{2} \le 6\) and \(x_{1}, x_{2} \ge 0\)
Then the Lagrangian function is :
Answer: A
The Lagrangian is \((10x_{1} - x_{2}^{2}) - \lambda_{1}g_{1} - \lambda_{2}g_{2}\), where \(g_{1} = x_{1} + x_{2} + r_{1}^{2} - 14\) and \(g_{2} = -x_{1} + x_{2} + r_{2}^{2} - 6\).
Each inequality \(g_{i}(x) \le b_{i}\) is turned into an equation by adding a squared slack variable, \(g_{i} + r_{i}^{2} = b_{i}\), which keeps the slack non-negative automatically. The Lagrangian for maximising \(f\) subtracts \(\lambda_{i}\) times each constraint written as \(g_{i} + r_{i}^{2} - b_{i} = 0\).
Option B uses slacks \(r_{i}\) without squares, which would need a separate sign restriction; option C rewrites the constraints wrongly; and option D changes the objective to \(10x_{1} + x_{2}^{2}\).
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