Discussion
A quadratic programming problem is given as follows :
Max. : \(Z_{x} = 2x_{1} + 3x_{2} - 2x_{1}^{2}\)
Subject to :
\(x_{1} + 4x_{2} \le 4\)
\(x_{1} + x_{2} \le 2\)
\(x_{1}, x_{2} \ge 0\).
If we apply Wolfe's method to solve, then the correct Kuhn-Tucker condition will be :
Answer: D
The conditions are \(4x_{1} = 2 - \lambda_{1} - \lambda_{2} + \mu_{1}\) and \(3 + \mu_{2} = 4\lambda_{1} + \lambda_{2}\).
Attach multipliers \(\lambda_{1}, \lambda_{2} \ge 0\) to the constraints \(x_{1} + 4x_{2} \le 4\) and \(x_{1} + x_{2} \le 2\), and \(\mu_{1}, \mu_{2} \ge 0\) to \(x_{1}, x_{2} \ge 0\). Stationarity of the Lagrangian for a maximum requires \(\nabla Z = \sum\lambda_{i}\nabla g_{i} - \mu\):
\[ \begin{aligned} 2 - 4x_{1} &= \lambda_{1} + \lambda_{2} - \mu_{1} \\ 3 &= 4\lambda_{1} + \lambda_{2} - \mu_{2} \end{aligned} \]
Rearranged, these are exactly option D. The other options attach the wrong coefficients: the objective has no \(x_{2}^{2}\) term, so no equation can contain \(x_{2}\) or \(4x_{2}\) (options A, B and C), and option C also has the wrong sign on the constant 2.
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