# Second-Order Conditions under Kuhn-Tucker Formulation

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?

• "...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. – Michael Nov 23 '19 at 23:05