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How should I address second-order conditions if I use the Kuhn-Tucker formulation of constrained optimization as opposed to the usual one?

For instance, suppose an agent wishes to maximize $f(x_1, x_2)$ subject to $x_1 \geq 0$ and $x_2 \geq 0$

In the Kuhn-Tucker formulation, the first-order constraints are:

$\frac{\partial f}{\partial x_1} \Big|_{(x_1, x_2)} \leq 0$

$x_1 \cdot \frac{\partial f}{\partial x_1} \Big|_{(x_1, x_2)}= 0$

$\frac{\partial f}{\partial x_2}\Big|_{(x_1, x_2)} \leq 0$

$x_2 \cdot \frac{\partial f}{\partial x_2} \Big|_{(x_1, x_2)}= 0$

But then how do you form the bordered Hessian to check the second-order constraints, since there aren't any constraint equations to differentiate? Are the second-order conditions the same, then, as an unconstrained problem?

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    $\begingroup$ "...as opposed to the usual one"---what is "the usual one"? "...since there aren't any constraint equations to differentiate?"---the constraints are right there in the problem, $x \geq 0$ is a constraint. $\endgroup$ – Michael Nov 23 '19 at 23:05

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