Discussion
Given the problem to maximize \(f(x)\), X = \((x_{1}, x_{2}, \ldots, x_{n})\) subject to m number of inequality constraints
\(g_{i}(x) \le b_{i},\ i = 1, 2, \ldots, m\)
including the non-negativity constraints \(x \ge 0\). Which of the following conditions is a Kuhn-Tucker necessary condition for a local maxima at \(\bar{x}\) ?
Answer: D
All of the listed conditions are Kuhn-Tucker necessary conditions for a local maximum, so the answer is All of these. For a problem of maximizing \(f(x)\) subject to \(g_{i}(x) \le b_{i}\), the conditions are: the stationarity condition, in which the partial derivatives of the Lagrangian with respect to each \(x_{j}\) vanish (option A); complementary slackness, \(\bar{\lambda}_{i}[g_{i}(\bar{X}) - b_{i}] = 0\) (option B); and feasibility of the point, \(g_{i}(\bar{X}) \le b_{i}\) (option C).
Because each of A, B and C is one of these necessary conditions, no single one of them is the only correct choice.
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